Halve avocado, remove pit, and scoop flesh into a medium bowl. Add lemon juice, 1/2 teaspoon salt, and 1/4 teaspoon pepper. Mash and stir with a fork until combined and mostly smooth, with some small chunks remaining. Divide avocado mixture among toast slices, spreading evenly. Top avocado with watercress, and drizzle with oil.
Bring 1 1/2 inches of water to a simmer in a large, deep skillet over medium-high heat. Stir in vinegar. Crack 1 egg into a small bowl. Holding bowl just above surface of water, gently pour egg into skillet. Repeat with remaining 2 eggs. Poach eggs, undisturbed, just until whites are firm and yolks are still runny, about 4 minutes (for firmer yolks, poach 1 to 2 minutes more).
Carefully remove eggs from water with a slotted spoon, allowing excess water to drain off. Place 1 egg on each watercress-topped toast slice. Season with salt and pepper. Serve immediately.