Preheat the broiler. In a large ovenproof pan over moderately high heat, warm 3 tablespoons of coconut oil.
Add the shiitake, garlic, and onion and stir until golden brown, about 8 minutes.
Stir in tamari and maple syrup and place under the broiler for 10 minutes, constantly tossing to evenly brown.
Spread remaining two teaspoons of coconut oil on the buns and toast. Assemble burger with avocado oil mayo, lettuce, tomato, shiitake "bacon" and ketchup. Serve with carrot sticks.