Step 1

Using your hands, crumble 1¼ lb. ground lamb into small pieces into a large bowl, scattering in a single layer across bottom and partway up sides of bowl. (This makes it easier to disperse the seasonings evenly without overworking the meat.) Using a Microplane, finely grate 1 lemon over meat; set lemon aside. Finely grate 2 garlic cloves into bowl. Sprinkle in 1 tsp. ground coriander, 1 tsp. Diamond Crystal or ½ tsp. Morton kosher salt, and ½ tsp. freshly ground black pepper.

Step 2

Finely chop ¼ cup (packed) cilantro leaves with tender stems and scatter half over lamb. Set remaining cilantro aside for making yogurt sauce. Toss ground lamb mixture with your hands just until combined—you don't want to overwork the meat mixture, otherwise it can become tough. Divide meat into 4 equal portions. Hold 1 portion in the palm of your hand, then gently pat down to make a patty that’s about ¾"-thick and just a bit wider than your buns (if you have small hands, it might be easier to do this on a cutting board). Place patty on a large plate and repeat with remaining portions.

Step 3

Spoon ½ cup plain whole-milk Greek yogurt into a measuring glass or small bowl. Cut reserved lemon in half, and squeeze juice from 1 half into yogurt. Add 2 Tbsp. extra-virgin olive oil, ¼ tsp. Diamond Crystal or Morton kosher salt, ¼ tsp. freshly ground black pepper, and reserved cilantro. Using a small whisk or fork, mix until oil is fully incorporated.

Step 4

Heat a large cast-iron or stainless-steel skillet over medium. Spread 4 Tbsp. room-temperature unsalted butter over cut sides of 4 hamburger buns, preferably whole wheat. Working in 2 batches, toast buns, buttered side down, in skillet until golden brown, about 3 minutes. Transfer to plates.

Step 5

Increase heat to medium-high and add 1 Tbsp. extra-virgin olive oil to skillet. Once oil begins to shimmer, arrange lamb patties in skillet and cook until a dark brown crust forms on bottom side, about 5 minutes. Using a spatula, flip burgers and cook until browned and an instant-read thermometer inserted into the center registers 135° for medium. Transfer to a plate and let rest at least 5 minutes.

Step 6

Place 4 cups baby arugula in a clean large bowl. Drizzle remaining 1 Tbsp. extra-virgin olive oil over, then squeeze juice from remaining lemon half over. Season with remaining ¼ tsp. Diamond Crystal or Morton kosher salt and ¼ tsp. freshly ground black pepper and toss to combine.

Step 7

Spread about 1 Tbsp. yogurt sauce on bottom half of each toasted bun. Top with a small handful of dressed arugula. Place patties on top of arugula. Spread 1 Tbsp. yogurt sauce over patties and close up hamburgers. Serve with remaining arugula salad alongside.