In a food processor, process both cheeses until broken into rough marble-sized pieces, about 10 seconds, then pulse until they have the texture of coarse sand, 5 to 10 pulses, scraping the bowl as needed. Transfer to a small bowl.

In the food processor, combine the pine nuts, garlic and ¾ teaspoon salt. Process until a smooth, peanut butter–like paste forms, about 1 minute, scraping the bowl as needed. Add the cheeses and about ½ of the oil and process until mostly smooth, 10 to 20 seconds, scraping the bowl as needed; the mixture should hold together when pressed against the bowl with a rubber spatula.

Using a chef's knife, roughly chop the basil, then add to the food processor. Pulse about 10 times, scraping the bowl several times, until the basil is finely chopped and well combined with the cheese mixture. Add the remaining oil and pulse just until incorporated, about 2 pulses. The pesto should be thick, creamy and spreadable.