Preheat the oven to 350°. Peel and slice the Onions into thin slices. In a heavy saucepan, put 3 tablespoons of the Olive Oil, the Thyme and the sliced Onions and season with Salt and Pepper to taste. Cook on medium heat until the Onions soften and start to caramelize. Stir occasionally and when brown, remove and lay out on a sheet to cool.
Slice the Mushrooms and sauté them in 1 tablespoon of Olive Oil for 5 minutes and then add the minced Shallots and Garlic. Cook another 5 minutes, stirring until the Mushrooms are cooked. Season with Salt and Pepper to taste.
Cut the Focaccia in two pieces, top and bottom, if thick enough to do so. Place the Focaccia halves on a baking sheet, inner part up, and cover with the cooked Onions and Mushrooms, covering each half of the Bread. Sprinkle the grated Cheese on the top. Bake until the Cheese starts to bubble (about 10 minutes). Cut into squares and serve with your favorite wine as an appetizer or entrée accompaniment.
Focaccia - An Italian staple probably originated with the Etruscans, this bread is similar in style and texture to Pizza dough, but much thicker. It's made from high-gluten flour, oil, water, salt and yeast. Bakers roll out or form it by hand into a round or, more often, square shape, about an inch or more thick and punctured with a knife to relieve bubbling during cooking in a stone oven. Variations abound all over Italy. Perhaps the most recognized is in Liguria in the northwestern bite below Piedmont, where Cheese Focaccia (Focaccia col Formaggio) is enjoyed in Genoa.