Bring a medium saucepan of water to a boil. Using a slotted spoon, gently submerge eggs into water. Simmer 5 minutes, then carefully immerse in cool water.
Heat 1 tbsp. butter with the oil in a 12-in. frying pan over medium-high heat. Add mushrooms and sauté until browned, about 8 minutes, then season with salt.
Whisk together crème fraîche, mustard, and lemon juice and zest in a small bowl. Gradually whisk in remaining 2 tbsp. butter to make hollandaise sauce.
Divide watercress among plates. Top with warm mushrooms, then drizzle with hollandaise. Crack an end of each egg and peel a quarter of shell. Using a spoon, carefully loosen eggs from their shells and scoop onto salads. Add a few turns of pepper on top.