Heat large nonstick skillet over medium-high heat. Add sausage and sauté until cooked through, about 10 minutes. Using slotted spoon, transfer sausage to bowl. Add onion, 1 tablespoon butter and thyme to skillet; stir 3 minutes. Add potato and bell pepper; season with salt and pepper. Reduce heat to low. Cover and cook until potato is tender, stirring occasionally, about 10 minutes.
Increase heat to medium-high. Stir sausage, beets and half of parsley into potato. Cook without stirring until hash begins to brown on bottom, about 5 minutes. Sprinkle cheese over. Remove from heat. Cover and let stand 5 minutes.
Melt 2 tablespoons butter in medium skillet over medium-high heat. Add eggs and fry to desired doneness.
Divide hash among plates; top with eggs. Sprinkle with remaining parsley.