Bring a large pot of salted water to a boil. Add the pasta and cook as the label directs for al dente, adding the asparagus during the last minute of cooking. Reserve 1 1/2 cups cooking water, then drain the pasta and asparagus.
Meanwhile, slice the corn kernels off the cobs. Run the back side of the knife down the cob over a bowl to remove any corn milk. You should get about 2 tablespoons.
Heat the olive oil in a large nonstick skillet over medium-high heat. Add the garlic, fennel, 1/4 teaspoon salt and a few grinds of pepper; cook until the fennel is browned in spots and tender, about 4 minutes. Add the corn kernels and corn milk and another 1/4 teaspoon salt and cook, stirring occasionally, until the corn is tender, about 2 minutes.
Add the pasta and 1 cup cooking water to the skillet and cook, stirring, about 1 minute. Remove from the heat and stir in the parsley, cheese, tarragon and butter until creamy, adding more cooking water as needed to loosen; season with salt and pepper.
Divide the pasta among bowls. Top with more cheese and pepper; drizzle with olive oil.