Heat 1 tablespoon of olive oil and 1 tablespoon of butter in a medium sized sauté pan over medium heat.
Add mushrooms and cook until tender and golden brown, stirring occasionally. Season with salt and pepper. Remove from pan.
To the same pan, add 1 tablespoon of olive oil and 1 tablespoon of butter. Add onions and cook until golden brown and caramelized, stirring frequently.
Remove from heat and add mushrooms back to the pan along with the thyme, balsamic vinegar, salt and pepper, stir to combine. Set aside and allow to cool (I put mine in the refrigerator).
In a large mixing bowl, sift flour, sugar, baking powder, baking soda, and salt. Cut in shortening and butter until mixture resembles coarse meal. In another bowl, combine egg and buttermilk and beat lightly with a fork.
Add buttermilk and egg to the flour mixture all at once and stir to make a soft dough. Add the cooled mushroom and onion mixture and blend together.
Turn dough out onto a floured work surface and knead 10 to 15 times. = Roll or pat dough into a square that is about 1-inch thick.
Cut dough into 9 squares (I like squares because then I don''t have excess scraps).
Place biscuits on a parchment lined baking sheet, brush with egg wash and sprinkle with coarse salt.
Bake at 425 for 12-15 minutes. Serve warm as an egg sandwich