Spread the beets on a baking sheet. In a bowl, whisk together the olive oil, vinegar, rosemary and some salt and pepper. Reserve one-quarter of the dressing for the salad and pour the rest over the beets. Roast until the beets are tender, 45 to 50 minutes.
While the beets are cooking, heat 1/4 inch of vegetable oil in a large skillet over medium heat.
Put the flour, eggs and breadcrumbs in separate shallow bowls. Coat the goat cheese slices first in the flour, then in the egg, then in the breadcrumbs. Fry the slices until golden brown, 1 to 2 minutes per side. Drain on a paper towel-lined plate.
Whisk the mustard and sugar into the reserved dressing, then pour over the greens in a large bowl and toss to coat. Divide the greens among 6 plates, add the beets on top and place 2 fried goat cheese slices on top.