Prepare an ice bath. In a small saucepan of salted boiling water, blanch the carrot tops for 30 to 45 seconds. Drain well, then transfer to the ice bath to preserve the green color. Shock for 5 minutes, then drain well and pat dry with a clean kitchen towel.
In a food processor or blender, combine the carrot tops with all of the ingredients except for the oil. We like to use all of the basil, including the stems. Puree until smooth. With the motor running, slowly stream in the oils until the pesto is thick and emulsified. If needed, add more oil. Season generously with salt and pepper.
Store in the refrigerator or serve right away with your favorite pasta, smeared on sandwiches, or as a sauce for grilled meat.