1. Melt 3 Tbsp. butter in a heavy 12-inch skillet over medium-high heat. Add potatoes, next 3 ingredients, and 3 Tbsp. water. Cook, stirring occasionally, 10 to 12 minutes or until potatoes begin to lightly brown.
2. Add mushrooms, thyme, and salt; cook, stirring occasionally, 6 to 8 minutes or until mushrooms are browned and tender. Add brisket, and cook, stirring occasionally, 6 to 8 minutes. Keep warm.
3. Melt remaining 1 Tbsp. butter in a large nonstick skillet over medium heat. Gently break eggs into hot skillet, and sprinkle with desired amount of salt and pepper. Cook 2 to 3 minutes on each side or to desired degree of doneness. Serve over hash.