Make the kale-mint pesto: in a food processor or high-speed blender add 4 cups tightly packed chopped green kale, 1 cup mint leaves, ½ cup walnuts, 3 large garlic cloves, and ¼ teaspoon salt. Pulse until the ingredients are coarsely chopped. With the machine running on medium speed, slowly pour in ¼ cup walnut oil until the mixture reaches a spreadable pesto consistency. Transfer the pesto to a bowl and set aside.
Shape the patties: divide 1 pound ground lamb into three equal portions and gently form into three even patties roughly the diameter of your hamburger buns.
Preheat the skillet: heat a large cast-iron (or heavy) skillet over medium heat until hot. Add 1 tablespoon grapeseed or olive oil and 1 tablespoon unsalted butter, swirling to melt and coat the pan.
Cook the burgers, first side: place the three patties in the skillet without overcrowding. Cook undisturbed for 5 minutes.
Cook the burgers, second side and melt cheese: flip the patties and cook an additional 3 minutes. Place one slice of jack cheese on each patty and cover the skillet (or reduce heat and cover) until the cheese melts and the meat is cooked to your desired doneness.
Toast the buns: while the burgers are cooking (for example, after you flip them), set your oven to broil. Place the 3 hamburger buns cut-side up on a baking sheet and broil, watching closely, until the cut sides are golden brown and crisp. Remove from the oven.
Assemble the burgers: spread the kale-mint pesto on the bottom half of each toasted bun, place a cheese-topped lamb patty on top, add a handful of spinach or spring green mix, then finish with the top bun.
Serve immediately.