Step 1

Season chicken with salt and pepper. Melt 2 Tbsp. butter in a large high-sided skillet over medium-high heat. Add leeks and half of lemon zest, season with salt and pepper, and mix to coat leeks in butter. Reduce heat to medium-low, cover, and cook, stirring occasionally, until leeks are somewhat tender, about 5 minutes. Remove lid, increase heat to medium-high, and cook, stirring occasionally, until tender and just starting to take on color, about 3 minutes. Add rice and cook, stirring often, 3 minutes, then add broth, scraping up any browned bits. Tuck short sides of each chicken thigh underneath so they are touching and nestle seam side down into rice mixture. Bring to a simmer. Cover, reduce heat to medium-low, and cook until rice is tender and chicken is cooked through, about 20 minutes. Remove from heat. Cut remaining 1 Tbsp. butter into small pieces and scatter over mixture. Re-cover and let sit 10 minutes.

Step 2

Meanwhile, pulse anchovy, garlic, capers, a few pinches of red pepper flakes, and remaining lemon zest in a food processor until finely chopped. Add herbs; process until a paste forms. With motor running, gradually stream in oil until loosened to a thick sauce. Add half of lemon juice; season salsa verde with salt.

Step 3

Drizzle remaining lemon juice over chicken and rice. Serve with salsa verde.