First, I preheated my oven to 375°F (190°C). Then I lightly greased a baking dish. Meanwhile, I seasoned the chicken breasts with salt and pepper, and heated the butter in a skillet. Once it melted, I added chicken and browned it on all sides—about 5 to 7 minutes—then transferred the chicken to my baking dish.
In the same skillet with remaining butter, I stirred in mushrooms, diced onion, garlic, and flour. I cooked that over medium‑high heat until everything softened, about 5 minutes. Then I poured in the chicken broth, brought it to a boil while scraping up the tasty bits from the bottom, and let it simmer until it thickened a bit.
Next, I drizzled the rich mushroom-onion sauce over the chicken in the baking dish. I baked it uncovered for about 30 minutes. Then I sprinkled the top evenly with the Parmesan, mozzarella, and diced scallions, and put it back in the oven for another ~10 minutes or until the cheese melted and the chicken was cooked through (I used a thermometer to check it reached 165°F / 74°C).