Move the top oven rack so it’s about 5 inches from the broiler element. Preheat the broiler and line a baking sheet with aluminum foil. Place the whole peppers on the sheet and broil, turning them every few minutes, until their skins are entirely charred. Transfer to a bowl large enough to hold them, and cover the bowl with the foil. Allow the peppers to rest for 20 minutes.

Uncover the bowl and use a paring knife to remove the skins, stems and seeds, while capturing the juice and flesh in the bowl. Cut or tear the peppers into 1-inch-wide strips. Using the flat side of a large knife, crush the garlic cloves and remove the skins. Add the garlic, olive oil and salt and pepper to taste and allow the roasted peppers to marinate at room temperature for 2 hours, stirring occasionally.

Preheat the oven to 350 F. If the chicken breasts are large, insert a knife with a 1-inch blade into the center from the end of the breast, creating a pocket. Be careful not to pierce the outside. If the breasts are small, butterfly them out flat, cover with plastic wrap, and pound flat.

Melt 2 tablespoons of the butter in a small sauté pan and briefly cook the mushrooms until they’re soft, 3 to 4 minutes. Fill the pocket of each chicken breast with a strip of pepper, then some of the mushroom, then the cheese, and repeat until it can hold no more. Close the opening with a toothpick, season to taste with salt and pepper, and dredge in the flour.

Melt the remaining 4 tablespoons butter in a large ovenproof sauté pan, add the olive oil, and cook the breasts, turning until lightly browned.

Transfer the pan to the oven and bake the chicken until cooked through, about 20 minutes. Serve immediately.