Preheat the oven to 400°F (200°C).
Halve each large chicken breast lengthwise to make four thin cutlets; season both sides with salt, pepper, and garlic powder, then lightly dredge each in flour, shaking off excess.
Heat the olive oil and 1 tablespoon butter in an oven-proof skillet over medium-high heat until hot.
Add the chicken cutlets and cook 4–5 minutes per side, until golden brown; transfer the cooked cutlets to a plate.
Add the remaining 2 tablespoons butter to the skillet and let it melt, then add the sliced cremini mushrooms and sauté, stirring occasionally, until browned, about 5–6 minutes.
Pour the Marsala and chicken broth into the skillet, bring to a gentle boil, and cook about 1 minute to combine flavors and reduce slightly.
Return the chicken to the skillet, spoon some sauce and mushrooms over each piece, and cook for about 1 minute to reheat.
Remove the skillet from heat, sprinkle the shredded mozzarella evenly over the chicken (concentrate on the chicken), and scatter the chopped scallions on top.
Place the skillet in the preheated oven and bake 7 minutes, until cheese is melted; optionally broil 1–2 minutes to brown the cheese. Serve immediately.
