Heat the oven to 450°F. Put a flat rack on a large rimmed baking sheet lined with foil.
With a meat pounder or a heavy skillet, lightly pound the chicken between two sheets of plastic wrap to even out the thickness of the breasts.
In a large bowl, mix 1/4 cup of the Parmigiano, 2-1/2 Tbs. of the capers, 1 Tbs. of the thyme, and the oil, mustard, lemon zest, salt, and pepper. Squeeze one or two of the lemon wedges to get 1 Tbs. juice and add to the mixture, along with 2 Tbs. water. Add the chicken and toss to coat well. You can proceed directly to coating the chicken breasts with crumbs, or let them marinate in the fridge for up to 24 hours.
Put the breadcrumbs in a large shallow dish and toss with the remaining 1/2 cup Parmigiano, 2-1/2 Tbs. capers, and 1 Tbs. thyme. Working with one piece at a time, transfer the chicken to the dish of crumbs, scoop some crumbs on top, and press well a couple of times so the crumbs adhere to both sides. Transfer to the rack on the baking sheet.
These crumbs are coarser than your average breadcrumbs, so it takes a little extra force to get them to adhere. My method is to pat a healthy amount of breadcrumbs on top of the chicken and then use the heel of my hand and a firm rotating motion to press the crumbs onto the chicken. Don't shake off the excess when transferring the breasts to the baking sheet; you want to keep as many crumbs on the chicken as possible.
Bake the chicken until it’s firm to the touch and registers 165°F on an instant-read thermometer, about 20 minutes. Serve immediately with the remaining lemon wedges for squeezing over the chicken.
Make the crumbs ahead. They'll keep for up to 3 days in an airtight container at room temperature. Or if you want to whip up a few batches of crumbs, you can store them in a zip-top bag in your freezer for a couple of months.
Marinate ahead. You can pound the chicken and immerse it in the sticky flavor bath the day before you plan to cook it. The results will be excellent because the flavors will really sink into the meat.