Heat cast-iron skillet or griddle for the burgers.
Heat a skillet over medium to medium-high heat. Melt butter and stir to melt or heat oil, whisk in flour and stir until deep golden and fragrant. Add chopped celery, bell pepper and onions, garlic, salt and pepper; cook 10 minutes. Add stock and bring to a low boil or bubble. Add shrimp and cook through, 2-3 minutes. Add hot sauce to your taste.
Place the turkey in a bowl. Add andouille to a food processor and pulse to finely chop. Add sausage to turkey with scallions and thyme, salt and pepper. Form 4 patties that are thicker at edges and thinner at the center. Drizzle patties with oil and cook 12-14 minutes to cook through.
Place lightly toasted bread on a plate, top with burger, cover with gravy.