Set oven. Preheat the oven to 375°F and adjust a rack to the center.
Make coating. In a large, shallow bowl mix the Parmesan cheese with 1 teaspoon of the oregano, 1 teaspoon of the thyme, and 2 teaspoons of the garlic. Set aside.
Add vegetables to pan. Coat a sheet pan with the olive oil and add the potatoes, mushrooms, onions, Poblanos, and the remaining 1 teaspoon of oregano, thyme and garlic. Use your hands to mix, being sure everything is well coated with the oil. Season generously with salt and pepper, and make four spaces for the chicken breasts to go -- they should not be on top of any vegetables.
Coat chicken. Place the chicken breasts on a clean, dry surface and season both sides generously with salt and pepper. Now, smooth side down, place them, one by one, into the Parmesan mixture and gently press down. Do this with each one, and add them to each cleared area on the sheet pan as you go. If not all of the Parmesan mixture has stuck to the chicken breasts, evenly divide what's remaining among them.
Bake. Place in the preheated 375°F oven and cook just until the chicken is cooked through, about 20 minutes. Remove the chicken breasts from the sheet pan, place them on a plate, and cover with foil. Turn the oven up to 425°F and continue to cook the vegetables until they're golden and slightly crispy, about 10 minutes.Place the chicken back on the sheet pan and place it under the broiler just long enough for the cheese to become golden, 30 seconds to 1 minute.