Cook the gnocchi according to package directions; drain.
Heat the olive oil in a skillet over medium heat. Add garlic to the skillet and cook for 2 minutes. Mix in the chopped onions and season with salt; cook until onions begin to soften, about 5 minutes. Stir in the mushrooms, peppers, tomatoes, and basil; cook for another 5 minutes. Add chicken stock with 2 tbsps white flour in it. Stir the butter in to melt. Pour the balsamic vinegar into the skillet, stir, reduce heat, and simmer the sauce for 15 to 20 minutes. Add shredded chicken. Toss the cooked gnocchi with the sauce.