Place the egg yolks, lemon juice, Parmesan, mustard, basil, garlic, 1 tablespoon salt, and 1 teaspoon pepper in a food processor fitted with the steel blade. Process for 20 seconds, until smooth. Combine the vegetable oil and olive oil in a 2-cup liquid measuring cup. With the processor running, slowly pour the oil mixture through the feed tube to make a thick emulsion. Taste for seasonings—the mayonnaise is a sauce so it should be highly seasoned. Store in the refrigerator until ready to use; it will keep for up to a week.