Heat the butter and 1 tbsp olive oil in a heavy skillet over medium low heat.
Add the mushrooms, garlic, and thyme and gently fry until the mushrooms have released their water and have become slightly browned. Stir often.
While the mushrooms are cooking, prepare a pot to poach your egg. For one egg a small pot will work. Fill it with water and a glug of white vinegar. The vinegar will help your egg to hold its shape when it hits the water.Heat the water until it is just on the edge of simmering, but not quite starting to bubble yet. When the mushrooms are done, set them aside but keep them warm.Add the other tbsp of oil to your skillet and fry your bread one side at a time, until both sides are golden. While you're frying your bread, poach your egg. Gently stir your nearly simmering water so that it's swirling. Crack your egg into a measuring cup. Pour the egg from the cup into the center of the swirl very gently. You might see some foam rise to the surface of the water, that's okay, you can just discard it. Allow your egg to cook for a few minutes, until the white is fully cooked but the yolk is still squishy feeling when you poke it.
Remove the egg to paper towel with a slotted spoon to drain slightly.
Spread your slices of avocado onto your bread, then top with the big pile of mushrooms. Lay your poached egg across the top and sprinkle with the feta cheese.