Fill a medium-sized saucepan with water and bring to a simmer. Gently slide each egg into the water and cook for 3 minutes or until the white has set and the yolk is cooked to your liking. Lift eggs out of the saucepan with a slotted spoon.
Meanwhile, heat 2 tbsp EVOO in a large frying pan. Add mushrooms and thyme, stirring occasionally. Cook for 10 minutes or untill golden and tender. Remove the pan from the heat and season with pepper and lemon juice.
Toast the bread and drizzle with remaining EVOO.
Assemble mushrooms and egg on toast and serve immediately.