Place the asparagus in a pot of lightly salted boiling water. Cook for 3 to 4 minutes, until tender-crisp. Drain. Run under cold water to stop the cooking.
Roughly chop asparagus and transfer to a food processor. Add the garlic, basil, pine nuts, and ½ teaspoon salt.
With the motor running, pour in 1 tablespoon of the oil and process until finely chopped. Set aside.
In a small bowl, combine the thyme, ½ teaspoon salt, ¼ teaspoon pepper, and the remaining tablespoon of oil. Rub the mixture over the steak.
Grill or broil the steak 4 to 5 minutes per side for medium. Remove the steak and let it sit, covered with foil, for 5 minutes.
Divide the steak into 4 portions and top with dollops of the pesto.