Add egg to a large bowl and gently whisk.
Add the rest of the hamburger Ingredients and mix with your hands until just barely combined.
Form mixture into 4 equal patties roughly the size of your hamburger buns. Using your thumb, make a deep impression in the middle of each patty. Set aside.
Meanwhile, in a small bowl whisk together all ingredients for the hoisin ketchup. Set aside in refrigerator.
In another small bowl, whisk together all the mayonnaise ingredients. Set aside in refrigerator.
Add coleslaw and carrots to a large bowl and toss with remaining slaw ingredients. Refrigerate until ready to use.
Heat 1 tablespoon olive oil in a large skillet over medium-high heat.
Once hot, add patties and cook until brown and slightly charred on one side, about 3-4 minutes. Flip and cook an additional 3-5 minutes. If you like your burgers more well done, add a couple minutes per side. If they start getting too charred before they're done cooking, lower the heat to avoid them burning.
While the burgers are cooking, spread each half of the buns with a little butter and toast.
When the burgers are done, let them rest about 5 minutes.
To assemble, spread the buns with some chili garlic mayo, top with the burger, top the burger with some hoisin ketchup and add a handful of the slaw.