Combine flaxseed and water in a blender or food processor and blend on high speed for 30 to 60 seconds until smooth. Transfer to a bowl and set aside.
Heat a large skillet over medium heat and add olive oil. Add garlic and onions, and cook until softened and golden brown, 2 minutes. Add tomato paste and mix well, reduce heat to low, and cook 5 minutes, or until the onions are cooked through.
Remove skillet from the heat and stir in lentils. Transfer to a large bowl. Fold in half of black beans and grated beet, mix well with a wooden spoon 2 to 3 minutes to combine.
Transfer half of this mixture into a blender or food processor. Pulse 15 to 30 seconds, until mostly smooth. Remove from blender or food processor and return to the bowl.
Add mushrooms, green onions, Worcestershire sauce and remaining black beans. Mix well with a spatula or with your hands for 2 to 3 minutes to combine. Add flaxseed mixture and continue mixing until all ingredients are combined.
Add oats and flour to bowl and mix another 2 to 3 minutes. Ingredients should have a sticky and coarse consistency. Cover and allow mixture to chill in fridge for 25 to 60 minutes.
Form ¼-inch-thick patties. Season with salt and pepper on both sides.
Heat an oiled flat skillet or indoor or outdoor grill over medium-high heat. Add patties and cook 3 to 5 minutes on each side.
Serve burgers on buns with your choice of toppings and condiments.