In a large bowl, combine all ingredients for the burgers and mix thoroughly.
Using a 1/3 cup measuring cup, scoop a ball into your hands (TIP: coat hands with a little olive oil to prevent sticking). Flatten ball into about a 1 1/2 inch thick patty and place on a nonstick baking sheet. Bake in oven for 30 minutes, flipping burgers about midway through. If desired, broil burgers to caramelize top after baking (optional). Keep warm until ready to serve.
In a large skillet, heat 1/4 cup sesame oil on medium high heat. When hot, add sliced mushrooms and allow to brown for 5-7 minutes, stirring occasionally. Midway through, add coconut aminos, garlic powder, and salt to taste. Keep warm on lowest heat for serving.
In a medium bowl, toss carrots, cucumber, vinegar, and honey. Place in refrigerator to keep cold.
Toast buns under broiler with olive oil (if desired), then assemble burgers with patty, shredded cucumber and carrot, sauce, mushrooms, fried egg, and kimchi (optional).