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Cook the fusilli in a large pot of boiling salted water until fully cooked, 10 to 12 minutes (you don't want al dente pasta here). Drain and toss into a bowl with the olive oil. Cool to room temperature.
In the bowl of a food processor fitted with a steel blade, purée the pesto, lemon juice, and 1/2 cup of the peas. Add the mayonnaise and purée until smooth.
Add the pesto-pea mixture to the cooled pasta, along with the Parmesan, 3/4 cup of the peas, 3 tablespoons of the pine nuts, the mozzarella, the salt, and the pepper. Mix well, then taste and adjust seasoning, if necessary. (I usually add 1/4 teaspoon each more salt and pepper, but it will depend on the saltiness of the pesto you're using and how heavily the pasta water was salted.) Transfer to a serving bowl and sprinkle the remaining peas, pine nuts, and basil over top. Serve at room temperature.
Note: To toast the pine nuts, place them in a small dry skillet over medium heat. Cook, stirring frequently, until golden. Watch the nuts carefully, as they can burn quickly, and transfer them to small bowl immediately after cooking (they will continue to brown in the hot pan).
Make-Ahead Instructions: The components of this pasta salad can be made ahead a day ahead of time; just keep everything separate and toss together right before serving.