Peel and roughly chop the onion and garlic. Add the onion and garlic to a food processor and pulse until roughly chopped. Add the portobello mushrooms and pulse until finely chopped, about 8 to 10 pulses.
Place a large nonstick skillet over medium-high heat with 1 tbsp vegetable oil. Once hot, add mushroom mixture and cook until onions are softened and mushrooms well browned about 5 to 7 minutes.
Transfer mushroom mixture to a medium bowl and wipe the skillet clean. Roughly chop the parsley leaves. Add parsley and cream cheese to the bowl with the mushroom mixture. Stir dumpling filling and season with salt and pepper.
Lay the dumpling wrappers out on a flat surface. Fill a small bowl with cold water. Use a tablespoon measure to scoop the dumpling filling and place in the center of each wrapper.
Dip your finger into the water, and wet the rim of the wrapper. Fold the wrappers over the filling and press down to seal, pinching the edge tightly.
Return the skillet to medium heat with 1 tbsp vegetable oil. Once hot, add the dumplings and cook until browned and crispy, about 2 to 3 minutes. Flip and cook dumplings an additional minute.
Add 1/3 cup water, reduce heat to medium, and cover. Steam the dumplings until tender and the water has evaporated, 3 to 4 minutes.
Thinly slice the apple and add to a large bowl. Add the Arcadian greens, champagne vinegar, 1 tbsp olive oil, and a good pinch of salt and pepper. Toss the apple arugula salad.
Divide the apple arugula salad and pierogi-style mushroom dumplings between large plates. Serve with sauerkraut and sour cream for dipping.