To prepare polenta, bring 4 cups water to a boil in a saucepan; stir in polenta and 1/2 teaspoon salt. Reduce heat, and cook until thick (about 5 minutes), stirring frequently. Keep warm.
To prepare ragout, cook carrot in boiling water 5 minutes or until crisp-tender; drain. Rinse with cold water; drain well.
Heat oil in a Dutch oven over medium heat. Add onion and garlic; cook 5 minutes or until soft. Stir in squash, 1/2 cup water, 1 tablespoon parsley, 1/2 teaspoon salt, and pepper; cover, reduce heat, and simmer 8 minutes or until squash is tender. Stir in carrot, 3/4 cup water, and peas. Bring to a simmer; cook for 10 minutes. Stir in 1 tablespoon parsley, basil, lemon juice, and butter. Serve ragout with polenta. Sprinkle with cheese.