Fill a large pot with lightly salted water and bring to a boil. Stir in penne, and return to a boil. Cook pasta uncovered, stirring occasionally, until mostly cooked through, but still firm to the bite, 8 to 10 minutes. Add shrimp to boiling water and cook until shrimp are bright pink on the outside and the meat is no longer transparent in the center, 2 to 3 minutes. Drain penne and shrimp; transfer back to empty pot.

Blend avocado, lemon juice, basil, 2 tablespoons Romano cheese, garlic, salt, and black pepper in a food processor until smooth. Add olive oil and pulse until creamy. Remove blade from food processor and stir diced tomatoes into the creamy pesto. Transfer pesto to pot with pasta and shrimp; stir to coat. Serve with about 2 tablespoons Romano cheese, salt, and black pepper.