Pulse mint in a food processor until finely chopped. Add shallot, lemon juice, cheese, 3/4 teaspoon salt, 1/2 teaspoon pepper, and oil, and blend until well combined, about 45 seconds. Add pistachios, and pulse until combined but still chunky. Refrigerate 30 minutes and up to 1 day; bring to room temperature before serving.
Bring a large pot of salted water to a boil. Add green beans; cook until barely tender and bright green, 2 to 3 minutes. Remove from water with a bamboo skimmer or a slotted spoon, and transfer to a bowl of ice water. Drain, and pat dry. Return water to a boil, and repeat with wax beans, then asparagus.
Arrange crudites on a large platter; top with half the sauce. Serve remaining sauce on the side.