Preheat oven to 375 degrees. Place the chicken breasts between two pieces of plastic wrap and pound thin.
In a sauté pan, drizzle with olive oil and then sauté the onion, garlic and mushrooms till tender and liquid is gone.
Add the spinach and cook just till wilted. Add salt and pepper to taste.
Remove the mixture from the stovetop and place in a medium sized bowl.
Add grated cheese and toss together.
Place 1/4 cup of cheesy veggie mixture to the center of the chicken breast and roll up.
Place seam side down in a casserole pan. Continue with the remaining chicken breasts. You will have some of the filling left over so add in the half and half and stir together.
Pour over the chicken and then grate asiago cheese over the top of the chicken breasts. Cook for 30-40 minutes.
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