Halve leek and slice ¼-inch thick; transfer to a large bowl and cover with cold water. Shimmy leek slices around in water then let rest 10 minutes. Using a slotted spoon or spider, scoop up leek slices, being careful not to kick up dirt at bottom of bowl, and transfer to a clean kitchen towel and pat dry.

Meanwhile, preheat a nonstick skillet over medium for 5 minutes. Add 1 teaspoon oil to preheated skillet and swirl to coat. Add English muffin cubes and season with salt and pepper; cook, stirring frequently, until golden brown and toasted, 10–12 minutes; transfer to a bowl.

Add 1 teaspoons oil to now-empty skillet over medium heat. Add leek slices and cook, stirring frequently, until softened and starting to brown, 3–4 minutes. Add kale and cook until just wilted, 1–2 minutes; season with salt and pepper. Transfer greens to a serving bowl. 

Wipe skillet clean then add remaining ½ teaspoon oil to skillet over medium heat. Crack eggs into skillet and cook until whites are set, about 2 minutes. Top each egg with 1 teaspoon Parmesan then carefully flip eggs, sprinkle another teaspoon Parmesan over each egg and cook 1 minute. Flip again and cook just until cheese is golden brown, about 1 minute. 

Top greens with eggs, croutons, avocado slices, fresh dill, flaky sea salt and black pepper.