Toss the zucchini on a rimmed baking sheet with the olive oil, 3/4 teaspoon salt and the crushed chili. Roast 20 minutes, or until the zucchini is golden brown on the edges.
Bring 4 or 5 quarts of water to a boil in a large pot and add 2 tablespoons salt. Add the pasta and cook until al dente.
Put the cream, shallot, garlic and lemon zest in a saucepan. Bring to a simmer over medium-high heat (watch that it doesn't boil over), then reduce the heat and simmer 10 minutes. Remove from the heat and stir the lemon juice, goat cheese, Parmesan.
Drain the pasta and transfer to a serving bowl. Add the chicken, cream mixture and zucchini and mix together. Scatter the basil over the pasta and serve hot.
For a vegetarian dinner, just omit the chicken. If you don't have cavatelli, shells or other small ridged pasta are other great pasta cuts you can use in this recipe.