Be sure to remove excess fatty skin from the thick ends of the breasts when trimming.
Total Time: 1¼ hours
1. Adjust oven rack to lower-middle position and heat oven to 325 degrees. Working with 1 breast at a time, use your fingers to carefully separate skin from meat. Peel back skin, leaving skin attached at top and bottom of breast and at ribs. Sprinkle 1½ teaspoons salt evenly over chicken (⅜ teaspoon per breast). Lay skin back in place. Using metal skewer or tip of paring knife, poke 6 to 8 holes in fat deposits in skin of each breast. Spray skin with oil spray.
2. Place chicken, skin side down, in 12-inch ovensafe skillet and set over medium-high heat. Cook, moving chicken as infrequently as possible, until skin is well browned, 7 to 9 minutes.
3. Carefully flip chicken and transfer skillet to oven. Roast until chicken registers 160 degrees, 25 to 30 minutes.
4. Transfer chicken to plate; do not discard liquid in skillet. Add garlic, ginger, oil, sugar, and remaining 1 teaspoon salt to skillet and cook over medium-high heat, stirring occasionally and scraping up any browned bits, until moisture has evaporated and mixture begins to sizzle, 2 to 4 minutes. Add broccoli and water and bring to simmer. Cover skillet, reduce heat to medium, and cook until broccoli is crisp-tender, 5 minutes, stirring halfway through cooking. Uncover and continue to cook, stirring frequently, until broccoli is fully tender and sauce coats broccoli, 2 to 4 minutes longer. Add any accumulated chicken juices to skillet and toss to combine. Season with salt to taste. Transfer broccoli to serving platter. Top with chicken and serve.