Make the kale-mint pesto: in a food processor or blender, combine the chopped kale (4 cups tightly packed), mint leaves (1 cup), walnuts (1/2 cup), garlic (3 large cloves), and salt (1/4 teaspoon). Pulse until the ingredients are coarsely chopped.
With the machine running on medium speed, slowly pour in the walnut oil (1/4 cup) and process until the mixture becomes a coarse pesto. Scrape down the sides as needed. Transfer the pesto to a bowl or container and set aside.
Divide the ground lamb (1 pound) into three even portions and shape each into a burger patty roughly the diameter of your buns; press gently to an even thickness.
Heat a large cast-iron skillet over medium heat until hot. Add the grapeseed or olive oil (1 tablespoon) and the unsalted butter (1 tablespoon) and let them melt and coat the pan.
Place the three patties in the hot skillet, leaving space between them. Cook undisturbed for 5 minutes. Flip the patties and cook an additional 3 minutes.
Place one slice of jack cheese on each patty and cover the skillet until the cheese melts and the meat reaches your desired doneness.
While the patties are cooking, set the oven to broil and place the hamburger buns (3), cut-side up, on a baking sheet. Broil until the cut sides are golden brown and crispy—watch closely and remove as soon as they reach the desired color.
Assemble the burgers: spread the kale-mint pesto on the cut sides of the toasted buns, place a cheesy lamb patty on each bottom bun, top each patty with a handful of spinach or spring green mix, and close with the top bun.
Serve immediately.