Line a plate with parchment or wax paper. In a large bowl, whisk together milk and bread crumbs. Add beef and salt and pepper to taste (I like 1 teaspoon salt and 1/4 teaspoon pepper) and knead until combined.
Using wet hands, shape the mixture into four 1/2-inch thick oval patties and transfer to lined plate. Chill at least 30 minutes or up to 4 hours.
In a large skillet over medium-high heat, melt 1 tablespoon butter until foaming. Add patties and cook until well browned, about 5 minutes per side. Transfer to a plate and reserve fat in skillet.
To the skillet, add onion and remaining 3 tablespoons butter and cook until softened, about 5 minutes. Add mushrooms, thyme, and 1/2 teaspoon salt and cook until the liquid has evaporated, about 5 to 7 minutes.
Stir in tomato paste and flour and cook until browned, about 2 minutes. Slowly stir in beef broth and red wine, and bring to a simmer.
Return patties to the skillet and cover. Simmer over medium-low heat until cooked through, about 12 to 15 minutes. Season the sauce to taste with salt and pepper (I don't usually anymore). Garnish with fresh thyme if desired.