How to strengthen pain de mie?

It calls for 8 ounces of water (1 cup = 8 fl oz = 8 0z weight of water, close enough) and somewhere in the neighborhood of 12-15 ounces of flour, depending on how one fills a cup with flour.  Dough texture could range from soft/tacky to stiff/dry.  Then it calls for a cup of rolled oats.  That definitely puts the dough into the dry range, which would explain your crumbly bread.  Plus, oats don't bring any gluten to the party.

I'd say convert the formula to weights, instead of volumes.  Even if you don't convert everything to weights, you will still need to increase the water content until you get a consistency that you like.  You could also soak the oats ahead of time, with either hot or cold water, so that they are moist already and don't pull water away from the dough.  The amount of water you put into the soaker will need to be factored into the total amount of water that you use in the dough so that you can understand how the finished dough is likely to behave.

If you want to stay with volume measurements, then try adding 1/4 cup more when you next make the bread.  Use it to soak the oats ahead of time.  As you make the bread, ask yourself how the dough feels with the additional water.  Is it sticking to everything it touches?  You might need to add a few sprinkles of flour to tame it and scale back the water for your next bake.  Is it moister than before but releases with gentle effort rather than sticking, tacky in other words, then it will probably work well.  If the finished bread is still drier than you want, then add another couple of tablespoons of water to the next bake.  Keep notes so that you know what the magic number is when you achieve it.

Have fun with your experimenting.

Paul

The recipe link is this one: Honey-Oat Pain de Mie. This is not for a standard (13" x 4") pain de mie pan (check the tip box)

As for the water amount, maybe try with the recipe in weight (ounce), rather than volume. Volume measurement is not universal.

Other recipes can be used for a standard (13" x 4") pain de mie pan, like Pain de Mie.

The first thing you need to do is get yourself a good kitchen scale and throw away the measuring cups; at least when you're making bread.  Your statement "I've shifted from AP flour to bread flour on one try. It was a bit tougher, but not much improved" tells us that you exchanged 3 cups of AP flour in the formula for 3 cups of "bread flour".  That means you didn't come anywhere near the same amount of flour (the error factor in using "cups" vs ounces or grams can be enourmous) so comparing one result to the other really has no value for you.

IMO the rolled oats contribute to the crumbliness of the bread, but it may have more to do with the length of time you are allowing for hydration before baking.  Rolled oats hydrate at a different rate than bread or AP flour.  Breaking down the ingredient amounts using standard weight conversions leads me to believe that the percentage of enriching ingredients appears to be pretty typical for pain de mie but if your bulk mesurements for flour were off by as little as half an ounce and just slilghtly for your water you'd be working with a formula that is bound to fail.

I use only weight measurements myself these days - I baked a lot of bread before home scales were common and don't find the answer to every question to contain "convert to weights."

Here are some points to consider:

1. How are you measuring the flour and the oats?  For both, it is best to use the "scoop and sweep" method.  Stir the flour lightly, then sppon it into the cup and used a flat blade to level it.  Don't shake or tap the cup.  This will give you the best approximation of the recipe writer's intent for the flour.
2. Have you followed the recipe?  It says one cup or more of water.  That's written for a reason.  My beloved bakers at King Arthur put it in terms of climate or season, but really, it has to do with many factors - you want to get the right dough texture (see below.)
3. Have you followed the recipe instructions?  A couple of things leap to mind here.  The instructions say to mix the dough and then let is rest for 20 minutes or so to hydrate the oats.  You want to be mixing to a fairly sticky dough and then let it rest before kneading.  Oats take up a lot of water and your dough will get easier to handle - then you want to further mix or knead the dough.  On this recipe, if I made an error, I would make the dough slightly wetter at the initial mix and if it still seemed too wet first let is sit longer and then add  flour during the final mix.  If you are making lean breads you should understand what a fully developed dough feels like.  You should pretty much mix this to a full development as it gets no folds during bulk fermentation.
4. On the recipe front - are you using a 9 inch pullman pan?  A poster above made a brief remark about this.  If you are mixing this recipe exactly as reproduced and then letting it final proof to fit a 13 inch pullman pan, you will have overproofed it.  It will be dry, open and (hmmm) crumbly after it is baked.  As posted above - see the tip...

Try working through these suggestions and see if they help.  King Arthur recipes are well tested, so you haven't started with a bad recipe - you just need to be careful in following the directions.

Hope this helps.

And yes, I think scales are a great baking tool (and not just for breads...) and formulas wirtten in weights make things easier to understand, but neither is required to make good bread.