Jammy Pepper Pasta Salad

Name: Jammy Pepper Pasta Salad
Ingredients: 1 cup nuts (such as hazelnuts, almonds, walnuts, or pistachios); 3 lb. mixed bell peppers (baby, regular-size, or a mix of both); 2 red Fresno chiles; 1 cup extra-virgin olive oil; 1 head of garlic, cloves separated; 3 large sprigs thyme; 2½ tsp. kosher salt, divided, plus more; 3 tsp. sugar, divided; ½ cup plus 3 Tbsp. red wine vinegar; ½ red onion; 1 lb. rigatoni; 6 oz. ricotta salata (salted dry ricotta), cut into very thin planks, or crumbled feta; 1 cup basil leaves