Drain the shiitake mushrooms, and add them to a food processor with the diced carrots. Pulse until everything is minced but not mushy. Alternatively, you can also grate the carrots, and chop the mushrooms finely with a knife.
To make the filling, heat sesame oil in a pan and add ginger and garlic to it. Saute for a few minutes till fragrant and add the carrot and shiitake mushrooms mixture. Cook the mixture for 4-5 minutes till the carrot is cooked through and add paneer, soy sauce, salt and pepper to the pan. Cook for another minute or so and set aside to cool.
To start wrapping the dumplings, prepare a large tray to keep the dumplings by spraying it with oil. Also keep a small bowl filled with water, the gyoza wrappers, the filling and a wet towel ready.
To wrap the dumplings, place the gyoza wrapper on your palm. Place the filling in the center and brush the edges with water. Start by bringing the center together and pinching it together with the index finger and thumb of one hand. With your other hand, start pleating one side of the wrapper onto the other till you have 2-3 pleats on either side. Pinch the edges together to seal the dumplings. (See video above to wrap dumplings)
Place them one by one on the greased tray, making sure they are not touching each other. Cover them with a wet towel to prevent them from drying out.
To cook the dumplings, heat a tablespoon of oil in a large skillet and place a few dumplings in the pan, making sure they don't overlap. When the dumplings start to sizzle, add 1/4 cup water to the pan and cover. Cook until all the water evaporates and the dumplings start to sizzle again (about 4-5 minutes). Remove them and repeat for the remaining dumplings. Serve immediately with a dipping sauce.