Bring a large pot of salted water to boil. Blanch broccolini for 2-3 minutes until bright green and crisp-tender. Drain and set aside.
Pat salmon fillets dry and season both sides generously with salt and pepper. Let come to room temperature for 10 minutes.
Heat olive oil in a large skillet over medium-high heat. When shimmering, add salmon skin-side up. Sear for 4-5 minutes without moving until golden brown crust forms.
Flip salmon and cook 3-4 minutes more for medium doneness. Remove to a plate and tent with foil to keep warm.
Reduce heat to medium and add 2 tablespoons butter to the same pan with the fond (browned bits).
Add minced garlic and sauté for 30-45 seconds until fragrant. Add capers with a splash of brine and let sizzle.
Pour in chicken broth and scrape up any browned bits. Let reduce for 1-2 minutes.
Add lemon zest and lemon juice. Stir in chopped parsley and season with salt and pepper to taste.
Add blanched broccolini to the pan and toss with the sauce for 1-2 minutes to heat through.
Swirl in remaining tablespoon of butter for extra richness if desired. Serve salmon immediately over the sauced broccolini.
Notes
Storage: Leftovers keep in refrigerator for up to 3 days. Reheat gently in a 300°F oven for 8-10 minutes. The sauce can be made ahead and gently rewarmed - whisk in cold butter at the end to restore silky texture.