Place the asparagus in a pot of lightly salted boiling water and cook for 3 to 4 minutes, until tender-crisp. Drain and run under cold water to stop the cooking. Cut the spears in half and remove the tops of the stalks. Roughly chop the stalk bottoms.
In a food processor, combine the chopped asparagus, garlic, basil, pine nuts, and 1 teaspoon of the salt. With the motor running, pour in 1 tablespoon of the olive oil and process until finely chopped. Reserve half the paste (about 1/2 cup) for another use.
In a small bowl, combine the thyme, pepper, and the remaining olive oil and salt. 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 steak into 4 portions and serve topped with dollops of Asparagus Tapenade.