Spread the chestnuts on a baking sheet. (To make peeling them easier, you can cut an X in the flat side of each chestnut with a sharp paring knife.) Roast in a 450ºF oven, turning every 5 minutes or so, for 15 minutes, or until the shells can be easily removed. Remove from oven and peel outer hull and the light brown inner skin. Chop the chestnuts roughly. Set aside.
Make the pesto by combining the pine nuts, basil, garlic, and salt in a large stone mortar and grind with the pestle until the moisture forms a paste. Slowly drizzle in the olive oil, beating all the while with a wooden spoon. Add the Parmigiano 1 tablespoon at a time, beating until the mixture forms a thick paste. The pesto can also be made in a food processor. Place in a large bowl and set aside.
In a medium bowl, add the raw zucchini slices and drizzle with about a tablespoon of olive oil and a pinch of sea salt. Once the zucchini slices are coated, add the pesto to the bowl with the raw zucchini.
Bring 6 quarts of water to a boil in a large pot, and add 2 tablespoons salt.
Cook trofie according to the package instructions. Drain and add to pesto and zucchini mixture; stir until thoroughly coated. Divide among four warmed plates. Sprinkle each dish with chestnuts and serve immediately.