In a large mixing bowl, combine the all-purpose flour and salt. Gradually add the warm water, mixing with a spoon or your hands until a shaggy dough forms. If the dough is too dry, add a tiny bit more water; if too wet, add a sprinkle of flour.

Transfer the dough to a lightly floured surface and knead for 8-10 minutes until smooth and elastic. It should feel firm but pliable. Form the dough into a ball, lightly oil the bowl, place the dough back in, cover with a damp cloth or plastic wrap, and let it rest at room temperature for at least 30 minutes. This allows the gluten to relax, making it easier to roll.

While the dough rests, prepare the filling. If not already pressed, press the extra-firm tofu for at least 15-20 minutes to remove excess water. Crumble the pressed tofu into a large bowl.

Finely chop the vegan kimchi and squeeze out as much liquid as possible. Add it to the crumbled tofu. Similarly, finely chop the napa cabbage, grate the carrot, mince the rehydrated shiitake mushrooms, slice the green onions, mince the garlic, and grate the ginger. Add all these prepared vegetables to the tofu-kimchi mixture.

To the filling bowl, add the soy sauce, toasted sesame oil, optional sugar, black pepper, and cornstarch. Mix everything thoroughly with your hands until well combined. The cornstarch helps bind the filling and absorb any remaining moisture. Taste and adjust seasonings if necessary.

Divide the rested dough into 2-3 equal portions. On a lightly floured surface, take one portion and roll it out thinly, to about 1/16 to 1/8 inch (1.5-3 mm) thickness. Use a 3-inch (7.5 cm) round cookie cutter or a glass to cut out dumpling wrappers. Gather the scraps, knead lightly, and re-roll. Keep unused dough and wrappers covered with plastic wrap to prevent drying.

Hold a wrapper in the palm of your hand. Place about 1-1.5 teaspoons of filling in the center. Lightly moisten the edge of half the wrapper with a little water using your finger.

Fold the wrapper in half to create a half-moon shape, pressing the edges together at the top center. Starting from one side of the center, create small pleats on the front side of the wrapper, pressing each pleat firmly against the back (unpleated) side. Repeat on the other side until the dumpling is sealed. Alternatively, you can simply seal the edges firmly with your fingers, or use a fork to create a crimped edge.

Place assembled dumplings on a parchment-lined tray, ensuring they don't touch, and cover them lightly with plastic wrap to prevent drying while you continue assembling.

Heat 1 tablespoon of vegetable oil in a large non-stick skillet or wok over medium-high heat. Once hot, carefully arrange a single layer of dumplings in the pan, ensuring they don't touch. Cook for 2-3 minutes until the bottoms are golden brown and crispy.

Carefully pour about 1/4 cup of water into the skillet. Immediately cover with a lid and reduce heat to medium. Let the dumplings steam for 5-7 minutes, or until the water has evaporated and the wrappers are translucent and cooked through. Remove the lid and cook for another minute or so to re-crisp the bottoms if needed.

Transfer the cooked dumplings to a serving platter and repeat with the remaining dumplings, adding more oil as needed. For steamed dumplings: arrange dumplings in a steamer basket lined with parchment paper (or lightly oiled), ensuring space between them. Steam over boiling water for 8-10 minutes, until wrappers are tender and translucent.

While dumplings are cooking, whisk together the soy sauce, rice vinegar, toasted sesame oil, and chili garlic sauce in a small bowl. Garnish with sliced green onions just before serving.

Serve the hot vegan dumplings immediately with the prepared dipping sauce.