1. Heat a large Dutch oven over medium-high heat. Add 1 teaspoon oil. Add spinach; sauté 2 minutes. Remove spinach from pan; drain, cool slightly, and squeeze out excess moisture. Add remaining oil to pan. Add garlic and shallots; sauté 3 minutes. Add sage, 1/2 teaspoon thyme, 1/4 teaspoon pepper, salt, and mushrooms; sauté 7 minutes. Stir in spinach, 1/2 cup walnuts, red wine vinegar, and cheese; cook 30 seconds.
2. Combine 8 cups water and white vinegar in a large saucepan, and bring to a simmer. Break each egg gently into pan. Cook 3 minutes. Remove eggs using a slotted spoon. Spoon 2/3 cup mushroom mixture onto each of 4 plates. Top each serving with 1 egg. Sprinkle evenly with remaining thyme, pepper, and walnuts.