Step 1

Thoroughly dry the chicken with paper towels. Generously season all over with salt and pepper.

Step 2

Heat the oil in a 10-inch or larger skillet over medium-high heat until shimmering. Working in batches if necessary, add the chicken and sear until deeply browned on the bottom, 5 to 7 minutes. Flip with tongs and sear the other side until browned, 5 to 7 minutes. Transfer the chicken to a plate; set aside.

Step 3

Reduce the heat to medium and melt the butter in the skillet. Add the mushrooms, onion, garlic, 1/2 teaspoon salt, and 1/4 teaspoon pepper and cook, stirring occasionally until tender, about 7 minutes. Add the flour, stir to evenly coat the vegetables, and cook for 1 minute.

Step 4

Stir in the half-and-half or milk and broth until no lumps from the flour remain. Bring to a boil. Reduce the heat to a simmer and stir in the Parmesan. Return the chicken to the pan and simmer until the sauce is thickened and the chicken is cooked through, 2 to 3 minutes. Sprinkle with additional Parmesan and the parsley before serving.