Add oil to pan over medium heat, add carrots and cook 5-8 min. Stir in garlic and tomato paste, cook about 1 min.
Add beef and ¼ tsp salt. Cook about 5 min. Stir in 1C water. Bring to rapid boil then reduce to medium and cook for 8-10 min. While sause thickens grate parmesan cheese and boil gnocchi 2-3 min. Be sure to break up any pieces that have stuck together.
Add butter and ½ the parm to the meat mixture and stir till butter is melted. Stir in gnocchi and reserved pasta water and cook till saause is thickened and glossy, about 1 min. Season to taste with salt and pepper.