In small bowl or mason jar, whisk together olive oil, lemon juice, shallot, mustard and salt.
Melt butter in a large cast iron skillet over medium-high heat. When butter's frothing subsides, crack eggs into the pan. Season with salt and pepper. Cook until whites are set, covering the pan for the last minute or so to firm up the whites on top.
In a large bowl, toss greens with enough dressing to lightly coat. Arrange on 4 plates. In same bowl, separately toss beets with a bit more dressing dressing and arrange on top of greens. Slide a fried egg on top of each salad.