Heat 1 tablespoon vegetable oil in a large skillet over medium-high heat.
Add the sausages and cook until browned on all sides, about 5–7 minutes. Transfer to a plate and set aside.
In the same skillet, add the onions, carrots, and celery. Cook, stirring occasionally, for 5–8 minutes, until the vegetables begin to soften.
Stir in the garlic and cook for 1 minute, until fragrant.
Sprinkle the flour over the vegetables and cook for 1 minute, stirring constantly.
Pour in the diced tomatoes and beef broth. Stir in the tomato paste and Italian seasoning. Season with salt and freshly ground black pepper.
Return the sausages to the skillet and stir to combine.
Transfer everything to a slow cooker. Cook on Low for 6–8 hours or on High for 4–5 hours, until the sausages are cooked through and the vegetables are tender.
For a thicker sauce, remove the slow-cooker lid for the last 30 minutes of cooking or stir in a cornstarch slurry (1 tablespoon cornstarch mixed with cold water).
Serve hot with mashed potatoes, crusty bread, or steamed green vegetables.