Put the beef in a large mixing bowl, and add all the marinade ingredients. Toss to make sure the beef is well coated. Cover with plastic wrap and refrigerate for at least 2 hours or overnight.
Preheat the oven to 375 degrees F.
In a medium bowl, mix together the tomato sauce and tomato paste until smooth. Stir in the kimchee liquid from the container, the oregano, parsley, garlic, paprika, fennel and red pepper flakes. Transfer the mixture to a saucepan, put over medium-low heat and simmer to blend flavors. Let simmer for about 20 minutes.
Preheat an indoor grill pan, but if not available, use a large skillet coated with sesame oil instead. Put a sheet of aluminum foil on the heated pan, and grill the beef, onion and mushrooms on the foil. Cook the beef until evenly browned, about 3 minutes on each side.
Roll out the pizza dough on a floured work surface, and transfer to a 17-inch baking sheet. Spread the dough with some of the pizza sauce in an even layer. Cut pieces of kimchee with scissors and sprinkle on top. Add some sliced fontina, and top with some meat. Repeat with the other piece of pizza dough. Put the baking sheets in the oven and bake for 15 to 20 minutes.
In a squeeze bottle mix the hoisin sauce, hot sauce and Korean red pepper paste.
While pizzas are still hot, sprinkle each with chopped green onions and drizzle with sauce from the squeeze bottle. Cut into squares and serve with a side of kimchee.
This recipe was provided by a chef, restaurant or culinary professional. It has not been tested for home use.