Season both sides of tilapia fillets with salt and black pepper. Heat oil in a large skillet over medium heat. Add tilapia fillets and cook 3 to 5 minutes.

Meanwhile, set a saucepan over medium-high heat and add the butter. Once melted, add the mushrooms, onions, garlic and thyme to the pan and cook, stirring often, until most of the liquid has evaporated, about 5 minutes.

Sprinkle the flour into the pan, stir to combine, and cook the roux, stirring, for 2 minutes. Whisk the milk into the saucepan. Season the sauce with the salt and pepper and stir to combine. Bring the sauce to a boil, reduce the heat to a simmer, and cook, stirring frequently with a wooden spoon, until sauce is thick and smooth and any floury taste is gone, about 5 minutes.

Top tilapia with mushroom mixture just before serving.