In a bowl, mix all of the ingredients for the dumpling sauce. Set this aside.
Season a skillet with a splash of sesame oil, and about 2 tsp of olive oil over medium high heat. Wait until the oil is shimmering, then add the mushrooms and ginger and garlic paste. Sauté for 2-3 minutes, until the mushrooms are mildly softened and appear to be a slightly darker brown.
Add half of the dumpling sauce into the pan, and continue to sauté the mushroom mixture until the sauce is fully cooked into the mushrooms.
Add the green beans to the pan, and sauté another 1-2 minutes. The green beans should be bright green and crisp. Remove the entire mixture from the pan and transfer to a bowl.
Fill a shallow bowl with room temperature water. Fill another bowl with 1/4 cup sesame seeds.
Dip a wonton or dumpling wrapper into the water, until soft to touch. Make sure the edges bend easily and that they are not dry.
Place the wrapper onto a clean, flat surface. Place 1 tsp of filling inside the wrapper. Pinch the edges and enclose the filling inside. If using a wonton wrapper, you will form a triangle when you fold and pinch the dough. If you are using a round dumpling wrapper, you will form a crescent half circle shape.
For wontons, bring the two outer corners of the triangle together, so they overlap. Basically, make the wonton "hug" itself. Push the middle of the two corners together until they stick together. For round dumplings, you can also gather all edges of the dough in the center, and twist until the filling is enclosed.
Flat side down, dip the wonton or dumpling into the sesame seeds. They should stick to the entire side.
Repeat for all of them.
In a skillet, heat olive oil over medium high heat, until it's shimmering. Sear the wonton/dumplings sesame seed side down for about 1 minute, until light golden brown. Flip and let them sit, until the other sides are also golden brown.
Remove from the heat and plate. Top with black sesame seeds and scallions. Serve with the rest of the dumpling sauce immediately.