Preheat oven to 350°F. Spray a 9x13-inch baking dish with cooking spray. If any chicken breasts are thick, pound them to an even, thinner thickness.
Melt 1/4 cup butter if not already melted.
Put 1/2 cup Italian breadcrumbs in a shallow dish. Brush both sides of each of the 6 chicken breasts with the melted butter, then press each breast into the breadcrumbs to coat both sides. Place the coated breasts in the prepared baking dish in a single layer.
Bake the chicken uncovered for 20 minutes.
While the chicken bakes, heat 1 teaspoon olive oil in a skillet over medium heat. Add 8 ounces sliced fresh mushrooms and sauté until they release their juices and most of the liquid has evaporated, about 5–8 minutes. Remove from heat.
After the initial 20 minutes of baking, remove the dish from the oven. Spoon 1/4 cup prepared basil pesto evenly over the 6 chicken breasts.
Divide the sautéed mushrooms evenly over the pesto-topped chicken, then place 1 slice provolone cheese on top of each breast (6 slices total).
Return the dish to the oven and bake an additional 10–20 minutes, until the chicken is cooked through and no longer pink in the center (safe internal temperature is 165°F) and the cheese is melted.
Remove from oven and let the chicken rest for about 5 minutes before serving.