Make the dipping sauce: Whisk the tamari, maple syrup, ginger, green onion, and sesame seeds together in a small bowl. Set aside.
Make the potstickers: Heat the olive oil in a large sauté pan over medium heat. Add the shallots. Stir and cook until fragrant and translucent, about 3 minutes. Add the shiitake mushrooms. Stir and sauté the mushrooms until they start to soften, about 2 minutes. Add the Brussels sprouts, garlic, and ginger, and stir. Season everything with salt and pepper. Keep stirring the filling until the Brussels sprouts are bright green and slightly wilted, about 1 minute. Remove from the heat, and allow the filling to cool slightly.
Set out a small bowl of water. To assemble the pot stickers, divide the vegetable filling among the wonton wrappers, placing about 1 tablespoon (15 mL) of the filling in the center of each wonton wrapper. Take one filled wonton wrapper and dip your finger in the bowl of water. Moisten two sides of the wrapper, fold all sides together, and pinch along the edge to form a seal. Repeat with the remaining filled wrappers.
Wipe the sauté pan and heat a thin slick of olive oil over medium heat. Fry the pot stickers in batches until they’re golden brown on all sides, about 1 full minute per side. Add more oil to the pan as needed to finish cooking all the pot stickers.
Serve the pot stickers hot with the dipping sauce on the side.