Preheat the oven to 350°F (175°C).

Spread half of the sliced mushrooms evenly over the bottom of a 9x13-inch baking dish.

Put the beaten eggs in one shallow dish and the Italian bread crumbs in a second shallow dish.

Pat the chicken breasts dry with paper towels. Dip each breast first into the beaten eggs, then into the bread crumbs, pressing the crumbs onto the chicken so they adhere.

In a 12-inch skillet, melt 2 tablespoons of the butter over medium heat.

Add half of the breaded chicken breasts to the skillet and cook until golden brown on both sides, about 4–5 minutes per side. Transfer the browned chicken to the baking dish, placing it on top of the mushrooms.

Add the remaining 2 tablespoons of butter to the skillet and brown the remaining breaded chicken the same way. Transfer the second batch of browned chicken to the baking dish on top of the first.

Sprinkle the remaining sliced mushrooms evenly over the chicken, then sprinkle the shredded mozzarella cheese evenly over the mushrooms and chicken.

Pour the chicken broth into the bottom of the baking dish (around the chicken, not necessarily over the cheese).

Bake the dish uncovered at 350°F until the cheese is melted and browned and the chicken is cooked through, about 30 minutes.