Heat oil in a large skillet over medium-high. Add shallots, thyme, and mushrooms; cook 4 minutes or until lightly browned. Add asparagus and garlic; cook 2 minutes. Add stock; cook 2 to 3 minutes or until liquid has reduced to about 2 tablespoons. Remove pan from heat; stir in butter, lemon juice, salt, and 1/4 teaspoon pepper.
Fill a skillet two-thirds full with water. Add vinegar; bring to a simmer. Break each whole egg into a custard cup, and pour each gently into pan; cook 3 minutes or until desired degree of doneness. Carefully remove eggs using a slotted spoon.
Divide mushroom and asparagus mixture evenly between 2 plates. Top each serving with 1 egg. Sprinkle remaining 1/4 teaspoon pepper over eggs.