Preheat oven to 400 degrees F with rack in the lower-middle position. Lightly oil a roasting pan and set aside.
Heat olive oil in a skillet over medium-high heat. Add mushrooms and shallots with 1/2 teaspoon each kosher salt and freshly-ground black pepper. Cook, stirring occasionally, until mushrooms are cooked down and starting to brown, about 5-8 minutes. Add garlic and cook for 1 minute, until fragrant.
Add a splash of white wine to the pan to deglaze, scraping up any brown bits on the bottom of the pan. Cook for an additional 1-2 minutes, until wine is evaporated. Stir in parsley and thyme and set aside to cool.
Gently loosen skin of chicken breasts by running your finger between the skin and the meat from the bone side outward. Be careful not to tear the skin or detach it completely from the chicken.
Add 1/4 of the cooked mushroom mixture to the pocket underneath the skin, spreading it evenly across the length of the chicken breast.
Add 1 ounce of goat cheese per chicken breast on top of the mushrooms and pull skin over the filling. If the skin appears quite loose, secure it with skewers to keep it closed over the filling.
Place stuffed chicken breasts in a single layer in the prepared roasting pan. Lightly brush the skin with olive oil and season with a pinch of salt and pepper.
Roast for 35-45 minutes (depending on the size of the chicken breast), until the internal temperature in the thickest part, away from the bone, reads 165 degrees F on a meat thermometer.
Remove from oven, tent with foil, and let rest for 10 minutes. Serve with a wedge of lemon.
Stuffing the Chicken Breasts:
Alternately, you can stuff the middle of the chicken breast. Insert a paring knife into the side and create an opening about 3 inches wide. Use the tip of the knife to cut through almost to the bone to create a pocket, being careful not to cut all the way through.Fill as you would under the chicken skin, and bake as directed.