Heat oven to 425. Dress asparagus with olive oil, salt and pepper. Roast for 15-20 minutes until charred at tips. Drizzle with lemon and season with salt and pepper to taste after removing from oven.
Meanwhile, saute shallot or garlic in butter for 30 seconds, then add mushrooms, thyme, plus salt and pepper to taste. Cook til liquid is evaporated and mushrooms are lightly browned. Add a splash of vermouth and deglaze pan. Again, cook til liquid is nearly evaporated.
Bring saucepan of water and splash of vinegar to boil and poach eggs, one at a time, 2-3 minutes each.
Split asparagus and mushrooms between 2 plates and top with a poached egg each. Season egg with salt and pepper.