Baking Measurements – Tips for Forming and Shaping Dough Uniformly

Damn, that looks good. I've never made this particular type of bread before, but here are some things that I do know which might help:

Butter is (roughly) 10% water and 90% fat, while lard is 100% fat. So if you substitute one for the other, you should adjust the amount of water in the recipe accordingly. The flavor will be a little different, but I bet either one would work.

I would not assume AP flour. In fact, I'd assume high-gluten bread flour. The gluten is what creates those fibers. It's the fat that keeps it tender.

This looks very similar to a brioche, so you might want to look at brioche recipes to learn some of the principles. One thing that most of them will tell you is to to knead the dough to develop the gluten fibers, and then add the butter. This I think is the key. I know that fat can keep gluten molecules from sticking together into strands, so if you add the fat later, you should wind up with strands of gluten surrounded by fat.

Here's why I think this is happening: In the oven, the internal temperature of the bread is going to transition from room temperature to cooked. The only way heat it introduced into the bread is at the surface. The sides have significantly more surface area than the center. The problem you are having is the sides are making this transition way before the center is. So the sides have very little time to rise and get cooked while the middle is in the prime rising temperature. So the yeast in the center has substantially longer to work before it is too hot for it.

You can try forming the bread thinner toward the center and thicker toward the edges. If you get the thickness right, it should be uniform.

I have also had good experience cooking on a stone. It will almost certainly speed up cooking time, but should help keep the temperature transition uniform across the whole loaf. So you don't end up with a thick part and a thin part; if you start out with a uniform thickness, you end up with a uniform thickness