Heat the vegetable oil in a medium saute pan over medium-high heat. Once the oil is hot, add the onions, garlic and ginger and cook until the onions are a deep golden brown, 5 to 7 minutes. Remove to a bowl. Add the shiitakes to the same pan and saute until browned, about 5 minutes. Add more oil to the pan, if needed. Add the cooked mushrooms to the bowl along with the edamame, pepitas and salt, to taste. Toss to thoroughly combine and adjust the seasoning. Allow to cool completely.
For the yuzu emulsion: Add the shallot, yuzu juice, Dijon, palm sugar and salt and pepper to a blender; blend until smooth. Drizzle in the olive oil until thick and smooth. Adjust the seasoning.
Add 1/3 cup filling to the middle of an egg roll wrapper with the longer side facing you. Wet the edge closest to you with egg wash or water and fold that side up over the filling. Wet the other edge furthest away from you and fold on top of the other edge, creating a seal. Fold the side pieces in towards the middle of the wrapper, resulting in a sealed rectangle. Set aside seam-side down as you repeat with the remaining filling and wrappers.
Heat 1/4 cup vegetable oil in a 14-inch cast-iron pan over medium-high heat. Once the oil is hot, add the bings to the pan in batches of 4, and cook until evenly golden brown, 3 to 4 minutes per side. Remove to a paper towel-lined plate.