Preheat the oven to 425 degrees F.
Peel and dice the sunchokes to 1/2-inch dice. (If you notice that the sunchokes are starting to oxidize and turn in color, cover them with water. When ready to roast them, strain and pat dry prior to tossing in salt and pepper.) Place in a small oven-safe roasting pan with 1 tablespoon olive oil, season with the salt and pepper and roast for 20 to 30 minutes, turning with a spatula after 15 minutes. Set aside to cool.
Cook the pasta according to package directions. Drain and toss well with the remaining 1 tablespoon olive oil. Cool to room temperature.
Add the pasta and the Roasted Red Bell Pepper Pesto to a large bowl and combine. Stir in the mozzarella, tomatoes, kalamata olives, green onions and the roasted sunchokes and serve.
In the bowl of a food processor fitted with the bottom blade, or in a blender, add the arugula, Parmesan, olives, vinegar, pepper, salt, garlic and roasted red bell peppers and process until combined well, 6 to 7 seconds. Slowly add in the olive oil and adjust the salt to taste. Refrigerate until ready to use.