To make the stuffing: Heat the butter in a heavy 9- or 10-inch skillet over medium-high heat. When the foam has subsided, add the diced mushrooms and chopped garlic; the skillet may seem quite full. Stir continuously for 1 min. as the mushrooms heat through and release their juices; they will shrink to fit in the pan. Spread the mixture evenly and cook for about 3 min., stirring once or twice to prevent sticking. As the liquid evaporates but before the pan becomes completely dry, add the pancetta and stir continuously for 1 or 2 min. Add the 1 Tbs. thyme and stir for 30 seconds. Transfer the stuffing to a bowl and stir in the Parmesan and a few grindings of pepper. Taste and add salt if needed (the pancetta and Parmesan may provide all the saltiness you need). Let the stuffing cool.
To make the chicken: Heat the oven to 400°F. Cut a pocket in each breast: Lay the breast skin side down on a cutting board. Hold a 5- or 6-inch sharp, thin-bladed knife with its flat side parallel to the cutting board. About half way up and midway along the thicker side of the breast, carefully make a 1-1/2- to 2-inch incision that goes deep into the breast but doesn't break through the opposite side. Pivot the knife forward to enlarge the pocket without widening the entry hole; remove the knife, reverse the blade, and then pivot in the opposite direction to complete the cavity. Be careful to avoid breaking through the surface of the breast.
Fill each cavity evenly with 2 Tbs. of the stuffing. Season the breasts with salt and pepper. Set a large, heavy frying pan over medium-high heat with 1 Tbs. butter and 1 Tbs. olive oil. When the fat is hot, cook the breasts (do this in batches) skin side down until golden, 4 to 5 min. Turn the breasts over and cook for 3 min. more. Transfer to an ovenproof dish large enough to hold all the breasts skin side up without touching; keep warm under foil while cooking the remaining breasts. Don't rinse the skillet; you'll need it for the sauce.
Put the chicken in the oven and roast until an instant-read thermometer inserted deeply into the meat reads 160°F, 10 to 15 min. The breasts will be plump and firm, yet will still yield a bit to the touch when cooked through.
To make the sauce: While the chicken is in the oven, remove all but 1 Tbs. of the fat from the skillet and set it over medium heat. Add the minced garlic and shallots and sauté until softened but not browned, 1 to 2 min. Add the verjus (or the grape juice and cider vinegar) and turn the heat to high. Scrape up the browned bits in the pan and reduce the liquid to about 2 Tbs., 3 to 4 min. Add the chicken broth and continue to cook on high heat until reduced to about 3/4 cup, 2 to 3 min. To thicken, add the flour and water mixture (through a sieve to prevent lumps) while stirring; add the 1/4 tsp. thyme and 1 Tbs. butter and cook for another 2 min., reducing the heat if necessary to slow the reduction. Taste for salt. Keep warm while finishing the chicken.
Cut the chicken breasts crosswise into 1-inch slices. Spoon 1 to 2 Tbs. of the sauce over each serving.