Cut mushrooms into small ½" dice. Trim ends off green beans. Stem, seed, and slice red bell pepper into thin ¼'' strips. Mince garlic. Cut twine into 3 equal lengths. Coarsely chop spinach. Rinse chicken breast and pat dry.
Make the Stuffing
Heat 2 tsp. olive oil in a medium pan over medium-high heat. Add mushrooms and cook 5 minutes, while stirring, until browned. Add spinach to pan and cook until wilted, about 2 minutes. Remove pan from heat and stir in half the asiago and season with ½ tsp. salt and ¼ tsp pepper. Allow to cool at least 5 minutes. Wipe pan clean.
Pound and Stuff Chicken
Place chicken breasts on a cutting board and cover with plastic wrap. Using a meat mallet, rolling pin, or small pan, pound chicken to an even ⅓” thickness. Remove plastic wrap and season both sides with salt and pepper. Lay a breast over 3 lengths of twine. Place stuffing oblong on chicken breast, leaving ½” border at edges. Place second breast over filling and secure by tying twine around stacked chicken breasts.
Cook Chicken and Vegetables
Heat same pan used for stuffing over medium-high heat. Add 2 tsp. olive oil and stuffed chicken to pan and cook 3-4 minutes on each side, or until golden brown. Transfer to one side of prepared baking sheet. No need to wipe pan clean. Add green beans, half the garlic, and red bell pepper to other side of baking sheet. Toss vegetables with 2 tsp. olive oil and a pinch of salt and pepper and roast 15-20 minutes, or until chicken reaches a minimum internal temperature of 165 degrees.
Make the Sauce
While chicken is resting, return pan used to sear chicken to medium-high heat. Place remaining garlic and Italian seasoning in pan and cook 30 seconds while stirring. Add tomato sauce and 3 Tbsp. water to pan and bring to simmer. Lower heat to medium and cook 2 minutes while stirring. Remove from heat and stir in remaining asiago and a pinch of salt and pepper.
Plate the Dish
Remove twine from chicken and cut into ¾” slices. Pour sauce on plate and arrange chicken slices over sauce. Place roasted veggies next to chicken.