Step 1

Stir lemon zest and juice, garlic, olive brine, red pepper flakes, half of tarragon, and 1 tsp. Diamond Crystal or ½ tsp. Morton kosher salt in a medium bowl.

Step 2

Score flesh side of chicken thighs with two or three ½"-deep slits. Add chicken to marinade and turn to coat. Let sit at room temperature 30 minutes, or cover and chill up to 3 hours.

Step 3

Place a rack in middle of oven; preheat to 425°. Heat oil in a large ovenproof skillet, preferably cast iron, over medium-high. Remove chicken from marinade, letting excess drip back into bowl, and place, skin side down, in skillet; season with salt. Cook, undisturbed, until skin is browned and crisp, 5–8 minutes. Turn chicken over, season other side with salt, and transfer to a plate.

Step 4

Add red onion, carrots, sumac, and remaining tarragon to skillet. Season with salt and pepper and stir to coat onion and carrots in fat. Return chicken to skillet, arranging skin side up. Transfer skillet to oven and roast until carrots are crisp-tender and chicken is cooked through (an instant-read thermometer inserted into the thickest part should register 165°), 20–25 minutes. Transfer to a platter and let rest 5–10 minutes.

Step 5

While the chicken is resting, toss red onion, lemon zest and juice, dates, pistachios, herbs, and olives in a medium bowl to combine. Season with salt.

Step 6

Spoon relish over chicken to serve.