Diced roasted beets (and crisp cucumber) are like colorful confetti on top of grilled asparagus. Roasting is a great way to cook beets; it concentrates juices, intensifies flavor, and yields tender roots.
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.
If you use two beets of different sizes,keep in mind that their cooking timesmay vary. Take each out of the oven asit's ready (knife-tender).