In a steamer insert set over boiling water, steam the carrots, fresh peas, and zucchini separately until each is tender but still firm, about 3 minutes for the carrots, 2 minutes for the peas, and 1 minute for the zucchini. Drain them and set aside. (It is not necessary to steam the frozen peas.)

In a large, deep skillet, heat the olive oil over medium heat. Add the scallions and sage leaves, reduce the heat to low, and cook, stirring, for 2 minutes. Add the chicken stock, fava beans, lentils, peperoncino, steamed vegetables, and frozen peas if using, bring to a simmer, and cook for 5 minutes. Season with salt and pepper to taste and remove from the heat. 

While you prepare the gnocchi, bring the salted water to a boil in a large pot over high heat. Cook and drain the gnocchi.

Meanwhile, bring the vegetable sauce to a simmer over low heat and check the seasoning.

Add the gnocchi and butter to the sauce and stir with a wooden spoon until the gnocchi are coated with sauce, 1 to 2 minutes. Transfer the gnocchi to shallow bowls, sprinkle with the grated cheese, and serve immediately. 

Note: For the prettiest dish, trim the zucchini and cut it lengthwise into quarters. Cut away the skin and about 1/4 inch of the white flesh in thick strips; discard the “seedy” center part. Cut these green and white strips into 1/4-inch dice.

For 1/4 cup cooked lentils, cook about 2 tablespoons lentils in a small saucepan of boiling water with 2 bay leaves until tender, about 20 minutes. Drain well.