Preheat oven to 400 degrees F.
Heat 1 tablespoon oil in a medium saute pan over medium heat. Add 1 red onion thinly sliced and saute until softened. Reduce heat to medium-low, season the onions with a pinch of salt, and continue to cook until caramelized. Add a splash of water if the pan is too dry. When the onions are caramelized, about 20 minutes later, add 1 tablespoon balsamic vinegar. Remove from heat and cool.
Place the mushrooms in a food processor and pulse until finely chopped. Wipe the saute pan clean and add the mushrooms. Season with ½ teaspoon salt and ½ teaspoon fresh thyme. Cook the mushrooms until all of the moisture has evaporated, stirring frequently. Remove from heat and cool.
Lightly dust a clean work counter with flour. Roll out the puff pastry into a 10x12 inch rectangle. Cut the rectangle into 6 rectangles, about 5x4 inches. Place a small amount of the mushrooms in the center of each, followed by the caramelized onions, 2 tablespoon fontina, and 2 ounces sausage.
Cut the pastry on both sides of the filling into ½ inch strips on a diagonal. Place the flaps on the short sides over the sausage mixture. Then take one strip over the filling from one side followed by another strip on the other side. Repeat to create a crisscross pattern.
Brush the sausage rolls with the whisked egg. Sprinkle extra fontina on top and sesame seeds. Bake the rolls for 25-30 minutes or until golden brown. Let rest for 10 minutes before serving.