Prep the vegetables by peeling and dicing the sweet potato, parsnip, and onion into small, uniform cubes.
Heat the ghee in a large cast-iron skillet over medium-high heat until shimmering.
Add the sweet potatoes and parsnips to the skillet, cooking for 8-10 minutes until they begin to soften and develop a golden-brown crust.
Stir in the diced onion and cook for another 3 minutes until translucent.
Add the ground venison to the pan, breaking it up with a wooden spoon, and season with thyme, sea salt, and black pepper.
Cook the venison until browned and cooked through, about 5-7 minutes, then stir in the minced garlic for the final 60 seconds.
Create a small well in the center of the hash and crack the egg into it, covering the pan for 2-3 minutes until the white is set but the yolk remains runny.
Garnish with freshly chopped parsley and serve immediately.