Heat 2 tablespoons olive oil in a large skillet over medium heat. Add mushrooms and sauté until most of the water is evaporated and they are golden. Add garlic and cook for 1 minute. Set aside to cool.

Combine ground beef, ginger, soy sauce, black pepper, and the cooled mushroom mixture in a bowl. Stir to combine and set aside.

Heat 2 tablespoons olive oil in a pot over medium heat. Add shallots and cook until softened and lightly browned, about 3-4 minutes. Add ginger and garlic to the pot and cook for 1 minute, stirring.

Add vegetable broth, soy sauce, and red pepper flakes to the pot. Bring to a boil, then reduce to a simmer. Cook uncovered for 10 minutes.

To assemble dumplings, place 1 tablespoon filling in the center of each wrapper. Moisten the edges with water and fold to seal. Repeat until all dumplings are assembled.

Heat 2 tablespoons olive oil in a large skillet over medium heat. Add dumplings in a single layer and cook until bottoms are golden, 2-3 minutes. Reduce heat to medium-low, add 1/4 cup water, and cover to steam until filling is cooked through, 5-7 minutes.

Transfer cooked dumplings to a plate and keep warm. Repeat for remaining dumplings.