In a large bowl, combine the kale and onion (reserving a few onion slices for topping the burgers, if desired) and toss with the lemon juice and 1 tablespoon olive oil; season with salt and pepper. Set aside, tossing occasionally.
Crumble the ground turkey into a medium bowl, season with salt and pepper and shape into four 3 1/2-inch patties, pressing your thumb into the center of each to form a slight indent.
In a large nonstick skillet, heat the remaining 2 tablespoons olive oil over medium-high heat. Add the patties and cook, turning once, until browned, about 8 minutes.
Meanwhile, toast the bread, then slather the cut sides with mayonnaise. Top each bread bottom with a burger patty. Using the same skillet, fry the eggs, then place 1 fried egg on top of each burger patty. Top with the reserved onion slices, if using, and set the bread tops into place. Serve with the kale salad.