For the grillades: Generously season beef with salt and pepper, lightly dredge in flour, and set aside. Heat 5 tbsp. of the oil in a large cast-iron or heavy pot over medium-high heat. Working in batches, brown meat well, 6–8 minutes per batch. Transfer meat to a bowl and set aside. Add the remaining 3 tbsp. oil to pot, then add onions, garlic, and peppers and cook, scraping up browned bits, until onions are soft, 8–10 minutes. Return meat and accumulated juices to pot. Add wine and bring to a boil. Reduce heat to low, cover, and simmer, stirring occasionally, until liquid thickens slightly, about 1 hour. Add tomato paste and cook until meat is very tender, about 3H hours more.
For the grits: Preheat oven to 375º. Bring 6 cups salted water to a boil in a medium pot over high heat. Gradually add grits, stirring constantly. Reduce heat to medium-low and simmer, stirring often, until thick, 10–15 minutes. Stir in 3 tbsp. of the butter, remove pot from heat, and set aside, stirring occasionally. Meanwhile, butter a large baking dish with 1 tbsp. of the butter; set aside. Whisk together milk and egg in a small bowl; set aside. Melt the remaining 3 tbsp. butter in a large skillet over medium heat. Add mushrooms and cook , stirring often, until soft, 4-5 minutes. Stir mushrooms and milk mixture into grits, season to taste with salt and pepper, and pour into prepared dish. Bake until warmed through, 15-20 minutes. Serve grits with grillades.