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Cut new potatoes into quarters and place in mixing bowl.
Place pine nuts, basil leaves, garlic, Parmesan, Pecorino and 1/2 cup olive oil in blender and grind until very fine (about 45 seconds). Add remaining oil in drizzle to form smooth yet very thick sauce and set aside. In another processor, place ceci, mint, lemon juice, egg, garlic and half cup of olive oil and blend until smooth, or 1 minute. Slowly drizzle in remaining oil to create a creamy sauce.
Dress the potatoes with half the pesto and place in center of one of 4 plates. Grill spiedini for 3 minutes per side and place one on each plate. Cover shrimp with remaining pesto and drizzle with aioli.