Bring a wide, large pot of water to a boil. Add the asparagus and cook 1 minute, until firm/tender. Drain well, cool asparagus in ice-cold water, drain well again, and pat dry.
(Asparagus can be made to this point hours in advance. Cover and refrigerate until ready to grill.)
Preheat your indoor or outdoor grill to medium-high. Set the asparagus on a baking sheet. Pour the 2 Tbsp. olive oil over the asparagus and roll to coat. Grill asparagus 1 to 2 minute per side, or until lightly charred. Arrange the asparagus on a platter. Drizzle with a little olive oil and balsamic vinegar. Sprinkle asparagus with the salt, pepper and basil and serve.