In a medium saucepan, combine the beet and enough water to cover. Bring to a boil, then simmer until tender, 10 to 12 minutes; drain.
In a large bowl, combine the beef, 1/3 cup parsley, the garlic and Worcestershire sauce; season with salt and pepper. Form into 4 patties and make a thumbprint in the center of each; coat the patties with the EVOO. Heat a skillet over medium-high heat. Add the patties and cook, turning once, 6 to 8 minutes for medium.
While the patties are cooking, crumble the goat cheese into a medium bowl. Add the sour cream, thyme, dill, lemon peel and the remaining 1 tablespoon parsley.
Place the beef patties on the roll bottoms; top with the beet slices, goat cheese mixture, onion, lettuce and the roll tops.