For mushrooms, carefully remove stems so as not to break the caps. Chop stems and reserve. With a small spoon, gently clean out and discard the gills from the mushroom caps.
Combine olive oil, balsamic vinegar, thyme and salt. Brush over both sides of mushroom caps and arrange rounded side up on a baking sheet. Bake in a preheated 400ûF (200ûC) oven for 10 minutes, until cooked through.
Meanwhile, for filling, heat remaining 1 tbsp (15 mL) oil in a skillet. Add garlic and onions and cook 5 minutes, or until tender but not brown.
Add chopped mushroom stems and cook until liquid evaporates and pan is dry. Add spinach, salt and dill.
Turn mushrooms cup side up. Fill each with the mushroom/spinach mixture. Sprinkle with cheese. Bake 5 minutes.
Bring a skillet of water to a boil. Add vinegar. Crack in eggs. Poach gently 3 to 4 minutes until whites are set and yolks are a little runny. Remove eggs from water to a plate lined with paper towels or a tea towel.
Trim away excess whites. Place eggs on mushrooms. Dust with pepper.