Line a plate with a paper towel and set aside. Heat a large nonstick skillet over medium-high heat. Fry the bacon, stirring occasionally, until crisp, 4 to 5 minutes. Remove the bacon to the lined plate with a slotted spoon or spatula.
Remove all but 2 tablespoons of grease from the skillet, reserving the remaining grease. Fry the eggs in the bacon fat for 2 to 3 minutes. Remove to a plate and cover with a paper towel to keep warm.
Add 1 more tablespoon bacon grease and turn up the heat to high. Add the sweet potatoes, parsnips, Brussels sprouts, onion, garlic and some salt and pepper, then cook, stirring occasionally, for 3 to 4 minutes. Deglaze the pan with 1/4 cup of water, then put the lid on and steam for 1 to 2 minutes.
Place fried eggs on top of hash and sprinkle over the bacon, diced tomatoes, goat cheese, parsley and basil.