Heat a large pot of salted water over medium-high heat and add the potatoes. Cook uncovered until the potatoes are slightly tender but not mushy, about 5 minutes after the water reaches a rolling simmer. Drain the potatoes and reserve on a paper-towel-lined sheet tray.
Heat 3 tablespoon of the olive oil in a large saute pan set over medium-low heat. Saute the onions until translucent, about 5 minutes. Reduce the heat to low, add the vinegar and continue to saute until the onions are slightly golden brown and caramelized, about 15 minutes, stirring occasionally.
In large bowl, whisk together the mustard, lemon juice, a pinch each salt and pepper and the caramelized onions. Slowly whisk in the remaining 1/4 cup oil to emulsify. Add the cooked potatoes and mix to coat. Taste and adjust the seasoning with salt and pepper if necessary. Add the eggs and gently stir, then serve.