Bring a large pot of salted water to a boil. Add the peas, asparagus, and green beans and cook until bright green and crisp tender, about 3 minutes. (If you're using frozen peas, you can throw them in during the last couple of minutes, to thaw.) Drain, transfer to a bowl of salted ice water to stop the cooking, and drain again. Puree the vegetables in a food processor with the ricotta, green onion greens, a drizzle of oil, and salt and pepper to taste.
Preheat the oven to 375 degrees F.
Brush the bread slices on 1 side with olive oil and put them, oiled sides up, on a baking sheet. Bake until lightly browned and crisp, about 10 minutes.
To serve, spread the puree on the warm croutons, and top with shaved Parmigiano (shave with a vegetable peeler). Serve immediately.