Heat the olive oil in a medium skillet over medium heat. Add the shallot and sauté until soft, 1 minute. Add the mushrooms and a generous pinch of salt, and sauté until soft and browned, 6 to 9 minutes, turning down the heat slightly, as needed.
Stir in the tamari, vinegar, and mirin. Stir, reduce the heat, then add the garlic, smoked paprika, and sriracha. Remove the pan from the heat and let cool slightly.
In a food processor, combine the sautéed mushrooms, walnuts, flaxseed, brown rice, and ½ cup of the panko. Pulse until just combined. The mixture should hold together when pinched, but it should still have some texture.
Transfer to a large bowl and fold in the remaining panko.
Form into 8 patties, place them on a large plate and chill in the fridge for 1 hour.
If you're grilling the patties, preheat a grill to medium-high heat. Brush the patties with olive oil and spray the grill with cooking spray. Place the patties on the grill and use a spatula to press down lightly. Grill for 7 minutes on the first side, flip, and grill for 6 to 7 minutes on the second side, or until well-charred and cooked through.
Alternately, cook the patties on the stove. Heat a cast-iron skillet over medium heat. Coat the bottom of the skillet with oil and cook the patties for 5 to 6 minutes per side, or until well-charred and cooked through.
Remove from the heat, brush with Worcestershire sauce, and serve with desired fixings.