In a large bowl, mix vinegar, Dijon, garlic, and salt with a whisk. Slowly drizzle in the olive oil while whisking vigorously.
Cut the frisée where the green part starts to turn white. Add the dressing and toss making sure all leaves are lightly coated. Set aside.
In a large fry pan over medium-high heat, melt the butter with the olive oil. Add the potatoes and thyme, and season with salt and pepper. Let potatoes fry, stirring often until they are browned, about 15 minutes. Add the shallots and cook until shallots are fried and potatoes are fork-tender, about 5 minutes.
Using the same frying pan for the potatoes, set over medium heat, add olive oil. Crack the eggs into the pan and fry to your preference. I like the whites fully cooked with a soft runny yolk, so I cook mine for about 4 minutes.
Put the salad onto the plates, spoon the warm potatoes over the salad. Top with the fried eggs and sprinkle a little salt and pepper and thyme.