Remove fat cap on lamb racks and trim well to remove all excess fat while being careful to leave a protective layer on meat. Strip rosemary from sprigs. Rub rosemary and lemon zest all over lamb racks; let marinate for 2 hours to overnight.
Roast walnuts in 350°F oven for 10 minutes until lightly toasted. In a mini food processor or by hand, finely chop 1-1/2 cups toasted walnuts; combine with 2 ounces of anchovies until well combined and very fine. Add the remaining nuts, anchovies and lemon. Add olive oil and combine.
Remove lamb from refrigerator 30 minutes prior to cooking. Remove most of the rosemary and all of the lemon. Season generously with salt and pepper. Grill racks over high heat until rare or medium-rare. Let meat rest for 10 minutes. Cut into individual chops.
Trim ends and stalks of broccolini. Toss whole broccolini pieces in olive oil. Season lightly with Kosher salt. Place broccolini on grill to obtain a nice char. If needed, place broccolini on a sheet tray and place in 350°F oven to finish cooking or for 3 to 4 minutes to reheat. Toss broccolini with peppers and roasted mushrooms.
For each serving, place broccolini and mushrooms on plate and drizzle with walnut anchovy salsa. Place 2 to 3 rib chops on each plate.