Heat chicken stock in a large pot over medium heat.
Heat olive oil in a skillet over medium-high heat. Saute onions until translucent, about 5 minutes. Transfer to the pot with the chicken stock; add garlic, basil, black pepper, and cayenne pepper.
Separate lobster shells and meat; place shells in the stock; cook until flavor is released, about 15 minutes. Remove shells and discard.
Place lobster meat in a blender; add eggs and Parmesan cheese. Puree until smooth. Place into the stock; add milk and whisk vigorously. Add pollock. Simmer until flavors meld, about 15 minutes.
