Place pork belly in a small pot with ½ cup vinegar, ½ cup water, bay leaves, salt, and pepper. Cover and bring to a boil over high heat. Reduce heat to low and continue to simmer for 1 hour. Drain and discard bay leaves. Let cool.
Preheat oven to 400 degrees F.
Place pork belly on a sheet pan and roast in the oven for 40 minutes, flipping over halfway through. Remove from oven and chop pork.
Melt butter in a medium saute pan over medium-high heat. Add red onion, ginger, garlic, and red chilies, and cook for 1 minute. Add pork, soy sauce, vinegar, and lemon juice, and cook until caramelized, about 2-3 minutes. Set aside and keep warm.
In the same pan, heat 1 tablespoon oil over medium heat. Add garlic and cook for 30 seconds or until golden brown. Add rice and cook until rice is hot, stirring occasionally. Season with salt and remove from heat.
Assemble the burritos. Spoon rice in the bottom ⅔ of the tortilla, followed by the sisig, fried egg, and cheese. Wrap and repeat with the remaining ingredients. Serve immediately with pico de gallo and salsa.