Place basil and garlic in a food processor. Grind into small pieces. Add chestnuts and cheese , salt and pepper and grind into thick paste. Slowly drizzle in olive oil to form smooth pesto.
Preheat oven to 350 degrees. Drizzle baking sheet with oil. Toss cut Brussels Sprouts in half of the pesto mixture and spread out in a single layer on prepared sheet.
Bake Brussels Sprouts for about 10 minutes, turn over sprouts and add remaining pesto. Continue roasting until sprouts are soft and golden brown, about 10 minutes. Turn as needed to prevent burning.