Heat oil in large saute pan and add onions.
Sprinkle with sugar and salt & pepper.
Stir to coat; cover and cook until onions are wilted and lightly golden. They should reduce by about 1/3 in volume. Allow to cool.
Mix cooled onions with mushrooms.
Preheat oven to 400 degrees F.
Place pie crust on parchment-lined baking sheet.
If necessary, drain any liquid from onion mixture before proceeding.
Spread about half of the onion mixture in the middle of the crust, staying about 1 inch way from the edge of the crust. Sprinkle with half of the cheese. Repeat with remaining onion mixture and cheese.
Fold edge of crust up to make a border around mixture. Do not cover the onions completely. Brush with egg mixture.
Bake for 30 - 35 minutes or until golden brown.
Remove from oven and allow to rest for 15 minutes.
Transfer to serving plate and slice into 8 wedges. Can be served warm or cold.
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