Heat oven to 350°F. Grease a 9x13-inch baking dish.
Brown the sausage in a skillet over medium heat. Drain and set aside.
Sauté onion, celery, and bell pepper in olive oil with 1 tbsp butter until tender, 6–8 minutes.
Stir in garlic, sage, and thyme; cook 1 minute until fragrant. Remove pan from heat.
Bring broth and milk to a simmer in a large pot. Whisk in grits slowly.
Cook, stirring often, until thick and creamy, 15–20 minutes. Take off the heat.
Stir in remaining butter, cheddar, salt, pepper, paprika, and hot sauce.
Temper the eggs: whisk eggs, then slowly whisk in 1 cup hot grits. Return mixture to the pot.
Fold in the sautéed vegetables, cooked sausage, and parsley until evenly combined.
Spread into the prepared dish and smooth the top. Bake 35–40 minutes until set and lightly browned.
Rest 10 minutes before scooping or slicing. Serve warm.