For the Umami Seasonings:
Combine the anchovies with the remaining ingredients to taste. Blend in a mortar and pestle or, for larger quantities in a blender or food processor. Set aside.

For the Umami Ketchup:
Purée the tomatoes with the juice from can in a blender until smooth. Cook the onion in oil in a heavy saucepan over moderate heat, stirring, until softened, about 8 minutes. Add the puréed tomatoes, tomato paste, brown sugar, vinegar, and salt and simmer, uncovered, stirring occasionally, until very thick, about 1 hour. Purée the ketchup in a blender until smooth. Chill, covered, overnight for flavors to develop. Then add the umami seasonings to taste and chill the ketchup until needed.

For the Oven-Dried Tomatoes:
Preheat the oven to its lowest temperature setting. Stir the brown sugar, tomato paste, soy sauce, and Worcestershire sauce together; brush on the sliced tomatoes. Put the tomatoes on a line sheet pan; dry in the oven overnight.

For the Caramelized Onions:
Cut the onions in half from pole to pole; peel and slice across the grain to ¼-inch thickness. Heat the butter and oil in a 12-inch nonstick skillet over high heat; when the foam subsides, stir in the salt and star anise. Add the onions and stir to coat; cook, stirring occasionally, until the onions begin to soften and release some moisture, about 5 minutes. Reduce the heat to medium and cook, stirring frequently, until the onions are deeply browned and slightly sticky, about 40 minutes longer.

For the Parmesan Crisps:
Preheat the oven to 375°F. Using the largest holes on a box grater, coarsely shred enough cheese to measure 1 cup. Line a large sheet pan with a nonstick liner, like Silpat. Arrange tablespoons of cheese 2 inches apart on the liner. Flatten each mound slightly with a spatula to form a 3-inch round. Bake in the middle of the oven until golden, about 10 minutes. Cool for 10 minutes on sheet on a rack; then carefully transfer each crisp with a metal spatula to a rack to cool completely.

To Assemble and Serve:
Grind the beef coarsely in a meat grinder or food processor. Put 6 ounces of meat into a 4-inch ring mold and gently tap down to form into a patty. Heat a cast iron skillet on high for 5 minutes. When it?s very hot, pour in a drop of vegetable oil to lubricate the pan. Season the patties liberally with salt and pepper. Add the patties to the skillet and sear on one side for 3 minutes; flip once and sear for 2 more minutes for medium rare.

In another skillet, add half of the butter and sauté the mushroom caps for until soft, about 2 minutes. Set aside. Remove the beef patties to rest. Wipe the mushroom skillet and toast the buns cut side down with the remaining butter.

Remove the buns when toasted and add spread about 2 tablespoons of the umami ketchup on both halves of the bun. Stack a beef patty with 1 tablespoon of the caramelized onions, a parmesan crisp, 2 mushroom caps and 2 slices of oven dried tomato. Serve immediately.