In a large pot cook pasta according to package directions until tender but still firm to the bite. Add asparagus to pot for the last 2 minutes of cooking. Drain pasta and asparagus. Rinse well in cold water; drain again. Transfer pasta and asparagus to a very large bowl.
Add olives, red pepper, copocollo, Fontina, cheddar, Gouda, and basil to pasta mixture. Toss gently to combine. Stir in dressing; mix well. Cover and chill for 2 to 24 hours before serving.