In a small bowl, mix together all of the ingredients for the herbed goat cheese. Refrigerate until ready to use.
Combine all ingredients for the lemon-dill yogurt sauce in a small bowl. Stir well and refrigerate until ready to use.
Wash, peel, and chop the medium beet and large carrots. Place them in a food processor and pulse into small rice-sized pieces.
Add the remaining ingredients for the burgers to the food processor. Turn it on and process until thick and sticky.
In a cast iron skillet, add enough coconut oil to generously coat the surface and heat it to medium.
Form burger patties out of the beet and rice mixture.
Place the patties on the hot cast iron skillet. Allow them to cook for 2 to 3 minutes or until crispy and golden.
Carefully flip the patties to the other side and let them cook another 2 to 3 minutes.
Serve the beet burgers on your choice of lettuce with the herbed goat cheese, thick slices of avocado, and a sprinkle of sprouts.