Put the potatoes in a large pot and cover with water by 3 inches. Add 2 tablespoons salt and bring to a boil. Cook the potatoes until they're tender when pierced, 20-25 minutes. Drain the potatoes and return them to the pot. Use the back of a ladle or large spoon to slightly crush the potatoes. Pour a tablespoon of olive oil over them, then cover to keep warm.
Slice the leeks in half vertically, and then into 1-inch wide half-moons. Put the leeks in a strainer and run under water to clean any grit or sand.
Put the leeks, 1 tablespoon of the olive oil and a pinch of salt in a medium cast iron or heavy nonstick skillet. Turn the heat to medium and cover the pan. Cook 10 minutes (checking the heat to make sure the leeks aren't browning) to steam and soften them. Remove the lid and continue cooking 5 minutes, until the leeks are tender and light golden.
Stir in the garlic and ramp leaves and cook until the leaves start to wilt - this should just take a few minutes. Add the leek mixture to the potatoes in the pot and stir together gently.
Wipe out the skillet and return to medium heat with a tablespoon of olive oil. Crack the eggs into the hot pan and sprinkle with salt. Immediately turn the heat to low and cover the skillet. Cook the eggs until done to your liking, 1 -3 minutes longer.
To serve, put the potato-leek mixture on plates or bowls. Sprinkle with some crumbles of goat cheese, then top with a fried egg. Season with black pepper to taste.