Preheat oven to 400 degrees. Wrap each beet in foil. Roast until tender when pierced with the tip of a knife, 45 minutes to 1 hour. When cool enough to handle, trim beets and rub off skins with paper towels, or peel beets with a paring knife.
Cut beets into 1/4-inch dice, and transfer to a medium bowl. Stir in cucumber, shallot, vinegar, 1 teaspoon oil, and 1/4 teaspoon salt. Refrigerate until ready to use, up to 2 hours.
Preheat a grill or grill pan until hot. Toss asparagus with remaining teaspoon oil and 1/2 teaspoon salt in a large bowl. Grill asparagus, turning once, until tender, 4 to 8 minutes (cooking time will vary depending on thickness of spears).
Arrange asparagus on a platter, and let cool completely. Finely chop basil, and stir into beet relish. Using a slotted spoon, spoon beet relish over asparagus.