Cook pasta in a large pot of salted water until al dente (the time the package stipulates is usually very accurate). Drain, reserving 3 tablespoons of pasta water for the skillet. Transfer the pasta to a bowl and toss with 2-3 tablespoons of the reserved brown butter.
While the pasta is cooking, add the red onion, garlic and jalapeño to a skillet with a few tablespoons of olive oil and lightly cook over medium heat for a minute. Add salt and pepper to taste and cook for 1 more minute.
Reserve a few asparagus tips and corn kernels for garnishing, and add the rest to the skillet.
Season with more salt and pepper if needed, increase the heat to medium/high and cook for 3 or 4 minutes, until everything is integrated and beginning to brown a little bit more.
Add the cooked pasta and the reserved pasta water to the skillet. Stir and then drizzle over the reserved brown butter. Carefully mix over low heat for a few seconds.
Remove from the heat, transfer to a serving platter, add chopped parsley and cheese, sprinkle with brown butter crumbs, and add the remaining asparagus tips and corn.
Serve immediately, passing more breadcrumbs and grated cheese if you want.