In a small bowl mash the lemon butter ingredients, including ¼ teaspoon salt and ¼ teaspoon pepper, until thoroughly blended.
Carefully lift the skin from each chicken breast, leaving the skin closest to the breastbone attached. Slide a paring knife under the skin and, without puncturing the skin, cut three to four slits in the flesh, each about ¾ inch deep. Spread 1 tablespoon of the lemon butter on the flesh of each chicken breast, pushing some of it into the slits. Lay the skin back in place.
Transfer the remaining lemon butter to a small saucepan and melt over medium heat. Set aside.
Prepare the grill for direct and indirect cooking over medium heat (350° to 450°F).
Remove and discard the tough bottom of each asparagus spear by grasping at each end and bending it gently until it snaps at its natural point of tenderness, usually about two-thirds of the way down the spear. Drizzle the asparagus with oil and season evenly with salt and pepper.
Brush the chicken breasts with oil and season evenly with salt and pepper. Grill the chicken, skin side up, over direct medium heat, with the lid closed, until nicely marked on the bottom, about 5 minutes. Without turning the chicken, move over indirect medium heat, close the lid, and continue cooking until the meat is opaque all the way to the bone, about 30 minutes more. About 5 minutes before the chicken is done, brush the tops with some of the melted lemon butter and then turn the breasts over, skin side down. Slide over direct medium heat, close the lid, and continue cooking until the skin is brown, basting once more with the lemon butter. Remove from the grill, cover, and let rest 3 to 5 minutes.
Grill the asparagus over direct medium heat, with the lid closed, until crisp-tender, 6 to 8 minutes, turning occasionally. Arrange the asparagus on a serving platter and drizzle with the remaining lemon butter. Serve the chicken warm with asparagus and garnish with lemon wedges.