For the tomato jam: Heat the oil in a pot over medium heat. Add the plum tomatoes, onion, salt and black pepper cook until the tomatoes are very soft and the onions are translucent, 20 to 30 minutes. Add the sugar, lime zest and juice, dried peppers, garlic, cumin, cinnamon and allspice and simmer for an additional 60 minutes.
Add the fire-roasted tomatoes to the mixture and puree. Refrigerate overnight.
For the Japanese bbq sauce: Whisk together the hoisin sauce, vinegar, honey and Sriracha in a small bowl.
For the onion strings: Heat the peanut oil to a temperature of 180 degrees F., using a high-heat thermometer for accuracy.
Whisk together the flour, cornstarch, salt, beer and cayenne if using in a large bowl to make the batter. In batches, dredge the onion in the batter, letting the excess drip off.
Cook the onions in the deep fryer in batches. Make sure you don¿t place too big a batch in the fryer or it will lower the oil temperature. Once the onions have started to brown a little, remove them from the deep fryer and place them onto a plate lined with a paper towel.
For the burgers: Fry the eggs sunny side-up on medium heat with a enough butter to keep the eggs from sticking, depending on the size of your pan and the number of eggs you choose to fry at once.
Cook the beef patties on a char broiler or grill to the desired temperature. Thirty seconds prior to taking the burgers off the char broiler, lay a piece of cheese on top of each and cover to melt.
Lightly toast the buns on the char broiler. Smear 1 1/2 ounces of the tomato jam on the bottom of each hamburger bun. Place a burger patty on top. Add a fried egg and then some of the onion strings. Drizzle each burger with 1 ounce of the Japanese bbq sauce to finish. Top with the bun tops. Reserve remaining tomato jam and bbq sauce for another use.
This recipe was provided by a chef, restaurant or culinary professional. It has not been tested for home use.