In a small jar with a lid, mix together vinegar, mustard, shallot, salt, and pepper, shake vigorously. Pour in the oil, shake vigorously to combine. Set aside while you cook the lentils and leeks.

Add six cups water, lentils, and salt to a medium saucepan with high sides. Bring to a boil, cover and reduce heat to low. Cook until just tender, about 20 to 30 minutes. Drain promptly and transfer to a large mixing bowl. Toss lentils with dressing.

Meanwhile, cut the leeks in half lengthwise almost all the way down to the root end. Wash the leeks very well under running water, making sure to rinse in between all the layers to remove any dirt. Cut off the root ends to separate the leek halves.

Arrange leeks cut side down in a single layer on the bottom of a 10 inch skillet. Add remaining three cups water, red wine vinegar, green onions, shallots, and parsley stems. Bring to a boil over high heat, cover with a lid or baking pan and reduce heat to low. Cook until leeks are tender when pierced with a fork, about 15 minutes. Line a plate with a few layers of paper towels. Use a spatula to transfer leak halves intact to the plate to drain excess liquid. Discard the outer layer of each leek along with any parsley stems, shallots, or scallions clinging to them. Keep leek layers stacked and cut leeks into 1/2 inch thick slices.

Reserve one tablespoon chopped parsley for garnish and mix remaining parsley in with the lentils. Divide lentils among four rimmed soup bowls. Arrange sliced leeks around the edges of the lentils, using one leek half per bowl. Heat olive oil in a large, preferably non stick, skillet over medium low heat. Carefully crack eggs into the pan and cook to desired doneness. Place one fried egg on top of each dish of lentils. Garnish with remaining chopped parsley.