Preheat the oven to 400 degrees F. Lightly coat 4 cups of a muffin tin with cooking spray.
Using a round cookie cutter, or the edge of a glass or small bowl (with a paring knife), cut out a large circle from each piece of bread.
Using your hands or a rolling pin, flatten the bread and place one flattened circle into each prepared muffin tin cup.
Heat the olive oil in a medium skillet set over medium-high heat. Add the mushrooms and bell peppers, and cook until softened, about 4 minutes. Season with salt and pepper.
Evenly divide the mushroom mixture between the four bread cups.
Working with one egg at a time, crack the egg over a bowl and let about half of the egg white fall into the bowl. Pour the yolk and the remaining egg white into one bread cup. Repeat with remaining eggs and bread cups. Reserve the egg whites in the bowl for another use.
Fill each empty muffin cup about a quarter full with cold water. This will stop the muffin tin from scorching in the oven.
Bake the egg cups until the white is set and the yolk is runny, about 10 minutes. This may vary by a few minutes.
Run a kitchen knife or small metal spatula around each cup, gently remove from the muffin tin.
Garnish with chopped parsley and hot sauce, if desired.