Grate beets and carrot in food processor and drain in a strainer set over a bowl. Press on vegetables to extract as much liquid as possible.
Add all other ingredients and form into 4 patties.
Preheat oven to 350F.
Over medium heat, add olive oil or ghee to a cast iron skillet or nonstick pan. I used a biscuit cutter set in the pan to mold the patties, but you can simply add your patties and sauté for approximately 3 minutes, until beginning to lightly crust on the bottom.
Transfer patties in the pan to oven and finish cooking for 20 - 25 minutes. Place patties on a lightly buttered bun or bagel, spread bottom with mustard, and top patties with sliced cucumber.