Heat grill for direct cooking over high, 450ºF–550°F (232ºC–288ºC). Brush grill grate clean and close lid to heat.
Generously coat all sides of salmon fillets with 1 ½ tablespoons oil and season tops generously with salt and pepper. Toss asparagus with remaining ½ tablespoon oil and season with salt and pepper.
Brush grill grate with oil.
Place salmon, top side down, and asparagus on grill grate, close lid, keeping temperature around 500ºF (260ºC). Cook salmon until ⅔ of the fillets are lighter in color, they’re becoming firm to touch, and they easily release from the grate, 4–6 minutes. Cook asparagus until charred and fork-tender, 6 minutes, turning halfway through.
Transfer asparagus to a platter. Flip fillets and generously brush tops of fillets with glaze, close lid and continue to cook until a thermometer inserted in the centers registers 130ºF (54ºC) for medium-rare, 2 minutes. (Cook to 145ºF for medium to well done) Transfer fillets to platter with asparagus.
Serve salmon with additional balsamic glaze and some crusty bread with good olive oil for dipping.