Make the Grillades: Gently pound the filet tips with a smooth meat mallet to approximately 1/4-inch thickness. Place the seasoned flour in a bowl. Sprinkle the cutlets with salt and pepper, dust them with the seasoned flour, and shake any excess back into the bowl. Reserve 1/4 cup of the seasoned flour.

Heat 1 tablespoon canola oil and 1/2 tablespoon butter in a heavy pot or Dutch oven over high heat. Working in 3 batches, saute the floured cutlets, turning once, until well browned, about 30 seconds per side. (Add some of the remaining 1 tablespoon canola oil and 1/2 tablespoon of butter between the second and third batches.) Transfer the grillades to a plate. Scrape the pot with a wooden spoon to remove any particles from the bottom.

Add the peppers, celery, onion, and garlic to the pot, and scrape the bottom with a wooden spoon. Season with salt and pepper, cover, and cook until tender and brown, about 8 minutes. Add the reserved seasoned flour to the vegetables and cook, stirring, until any liquid is absorbed and the vegetables start turning a darker brown, about 1 1/2 minutes. Scrape the bottom of the pot with a wooden spoon to cook the flour and give the sauce some color. Add the wine, lower the heat, and simmer for 2 minutes.

Gradually stir in the stock, again scraping the bottom of the pan with a wooden spoon, and bring to a boil. Add the tomatoes, bring to a boil again, and reduce to a slow simmer. Simmer until the sauce is thick enough to coat the back of the spoon, about 30 to 45 minutes. Season with salt and pepper, to taste.

Make the Grits: Heat the butter in a large saucepan over medium heat. Add the onion and cook, stirring, until translucent, about 4 minutes. Add the milk and season with salt and pepper, to taste. Bring to a simmer while stirring occasionally. Add the grits, stir thoroughly to blend with the milk, and boil for about 2 minutes. Reduce the heat and simmer, stirring frequently, until the grits thicken, about 20 to 30 minutes. Stir in the thyme. Cover, turn off the heat, and let rest for 10 minutes. Season with salt and pepper, to taste.

To serve, place 4 pieces of the grillade on each plate spoon some of the vegetables and sauce over the top. Sprinkle with green onions and spoon a portion of grits in center of each plate.