Heat 2 tablespoons oil in a large skillet, preferably cast iron, over high heat. As soon as oil starts to smoke, add shishitos to cover bottom of pan in a single layer. Don’t crowd. Cook about 3 minutes, tossing occasionally, until blistered in spots but still green. (You don’t want them to collapse or soften too much.) Let sit until cool enough to touch. Coarsely chop peppers and place in a small bowl. Mix in grated garlic, shallot, vinegar and 4 tablespoons oil; season with salt and set aside.
In a large bowl, combine the lamb, 2 tablespoons minced garlic, 2 teaspoons salt and 2 teaspoons cracked pepper. Form the mixture into small, thin patties.
Heat the same cast-iron skillet over high heat. Add the lamb burgers and cook for about 2 to 3 minutes on each side for medium. Remove lamb and add kale leaves to pan. Sear until just wilted and slightly brown. Place each lamb pattie on a kale leaf, top with shishito relish, wrap in leaf and serve.