Heat 2 tablespoons oil in a large high-sided skillet over high heat. Cook mushrooms, stirring occasionally, until browned, about 7 minutes. Add leeks and 1/2 teaspoon salt. Cook for 2 minutes.
Add stock. Reduce heat to medium. Simmer until reduced by half, about 5 minutes.
Meanwhile, bring a large pot of water to a boil. Cook pasta according to package instructions until al dente. Drain, reserving 1/4 cup cooking water.
Add pasta, reserved cooking water, and 1 cup asparagus curls to skillet. Toss to coat, and season with pepper.
Divide pasta evenly among 4 plates. Top with remaining cup asparagus; drizzle with remaining 2 teaspoons oil. Top with cheese.