Heat 1 tablespoon oil in a large nonstick skillet over medium-high heat.
Add veggie burgers and cook, turning occasionally, until browned, 5 to 7 minutes. Transfer to a clean cutting board; dice when cool.
Heat 1 tablespoon oil in the pan over medium heat. Add sweet potato and onion and cook, stirring occasionally, until browned and tender, 13 to 15 minutes, reducing heat as necessary.
Stir in kale and water and cook, stirring, until wilted, about 2 minutes. Add garlic and thyme; cook, stirring, for 30 seconds. Stir in the diced veggie burgers, salt and pepper. Cook, stirring for 1 minute. Transfer to a bowl and cover to keep warm.
Meanwhile, mix mayonnaise, ketchup and hot sauce in a small bowl.
Wipe out the pan and heat the remaining 1 tablespoon oil over medium heat. Cook eggs, without flipping, for 3 to 4 minutes for sunny-side up.
Serve the hash and eggs with the sauce. Garnish with more pepper and thyme, if desired.