Combine the avocado and salsa in a small bowl; set aside. Whisk 2 3/4 cups water, the grits, 1 teaspoon salt and a few grinds of pepper in a microwave-safe bowl. Cover with plastic wrap and pierce a few times with a knife; microwave until thickened, 8 minutes. Uncover and microwave until creamy, 4 to 5 more minutes. Whisk in the cheese. Cover and set aside.
Meanwhile, form the beef into four 3/4-inch-thick patties. Heat a large cast-iron skillet over medium-high heat; sprinkle the skillet with1 teaspoon salt. Add the patties and cook 4 to 5 minutes per side for medium doneness. Transfer to a plate and tent with foil.
Add the scallions and jalapenos to the skillet. Cook, stirring, until tender, 2 minutes. Add the kale; stir until wilted. Transfer to a bowl.
Wipe out the skillet. Add the butter and swirl to melt. Add the eggs and fry as desired; set on top of the burgers. Whisk the grits, adding hot water to loosen, if necessary. Divide among plates and top with the kale mixture and avocado salsa. Serve with the burgers.