1. Bring 4 quarts water to boil in stockpot. Add 1 tablespoon salt and pasta, stir to separate, and cook until al dente. Drain and return to pot.
2. Just before putting pasta in boiling water, bring balsamic vinegar to boil in small skillet over medium-high heat; reduce heat to medium and simmer slowly until syrupy and reduced to 2 tablespoons, 5 to 10 minutes.
3. While pasta is cooking and balsamic is reducing, heat 1 tablespoon oil in 12-inch nonstick skillet over high heat until beginning to smoke. Add asparagus, onion, black pepper, pepper flakes, and remaining 1/4 teaspoon salt and stir to combine; cook, without stirring, until asparagus begins to brown, about 1 minute, then stir and continue to cook, stirring occasionally, until asparagus is tender-crisp, about 4 minutes longer.
4. Add asparagus mixture, basil, lemon juice, 1/4 cup Pecorino, and remaining 2 tablespoons oil to pasta in stockpot; toss to combine. Serve immediately, drizzling 2-3 teaspoons balsamic glaze over individual servings and passing remaining 1/4 cup Pecorino separately.