Cook the bacon in an oiled pan over medium-high heat until golden and crisp, then set aside, leaving the drippings in the pan.
Add shallots, carrot, and celery; sauté for 5–6 minutes until softened, adding a splash of water or stock if needed.
Stir in garlic for 30 seconds, then add ground beef and cook until browned.
Return the bacon and mix in dried herbs, tomatoes, tomato paste, balsamic vinegar, sweetener, and stock. Bring to a boil, reduce heat, cover, and simmer for 20 minutes until thickened.
Add the gnocchi and simmer uncovered for 5–8 minutes until tender, adding more stock if the sauce thickens too much.
Season with salt and pepper. Transfer to a baking dish, top with grated cheddar, and broil for 4 minutes until the cheese melts and bubbles.
Garnish with fresh basil and serve hot.