In a large skillet, cook bacon over medium heat, stirring, until crisp and browned, 8 to 10 minutes. Using a slotted spoon, transfer bacon to a plate. Pour off all but 1 1/2 Tbsp. of fat and add onion. Cover and cook over medium heat, stirring occasionally, until softened, about 5 minutes. Uncover and cook, stirring, until lightly golden, about 2 minutes longer. Working in batches, add kale; season with pepper. Stir in 1 Tbsp. water and cook, covered, over medium-high heat, stirring occasionally, until kale is wilted and tender, about 5 minutes. Return bacon to skillet.
In a large nonstick skillet, warm 1 Tbsp. oil over medium heat. Add eggs, cover and cook for 2 minutes. Flip and cook for 10 to 30 seconds longer, depending on desired level of runniness.
Brush toasted English muffins with remaining oil and rub lightly with cut side of garlic clove. Mound greens on bottom halves of muffins and top each with an egg. Close sandwiches and serve right away.