Cut peppers in half through stem; remove seeds and veins.
Bring about 4 quarts of water to a boil. Add pepper halves and boil for 5-7 minutes. This blanches the peppers, but the peppers should not get soft. Remove and drain. Set aside to cool.
Slice chicken breast into 1/4-inch slices. Dredge in seasoned flour.
Heat a large skillet over medium-high heat. Add oil and butter (it’s OK if the butter browns–it adds flavor). Add the chicken and brown.
Add the mushrooms, onions, garlic and thyme. Saute for 5 minutes.
Add the stock packet (or cube) and the wine (or water). Stir; reduce heat to low, cover and cook for 10 minutes.
Add the cream and the shredded asiago cheese, stir until cheese is melted.
Spoon chicken mixture into pepper halves. Top with mozzarella slices.
Heat broiler to HI heat.
Place peppers on baking sheet and broil until cheese is melted, bubbly and begins to brown.