Start the night before or early morning by washing the red mung beans until the water runs clear. Soak them in plenty of cold water for 4-6 hours or overnight.
After soaking, drain the beans and rinse them one more time. Put them in a pot with 8 cups of fresh water. Turn the heat to high and bring to a boil. Once boiling, lower the heat to medium-low and cook for about 30-40 minutes. You'll know the beans are ready when you can easily squish them between your fingers. Drain all the water using a strainer.
Put the cooked beans back in the pot. Add the vegetable oil, sugar, and salt. Cook over medium-low heat, stirring constantly with a wooden spoon. Keep cooking and stirring until the mixture becomes very thick, about 15-20 minutes. You'll know it's ready when you can draw a line through the mixture with your spoon and the line stays clear. Put this filling in a bowl and let it cool completely, about 1 hour.
While the filling cools, make your dough. Mix the flour and salt in a large bowl. Add the room temperature lard in small pieces. Use your fingertips to work the lard into the flour until it looks like sand. Slowly add the room temperature water while mixing with your hand. Knead gently for 3-4 minutes until you have a smooth dough that doesn't stick to your hands. Cover the dough with plastic wrap and let it rest for 30 minutes on your counter.
Now preheat your oven to 350°F (180°C). Line your baking sheets with parchment paper. Mix the egg yolk, milk, and pinch of salt in a small bowl for your egg wash.
Divide your cooled filling into 20 equal portions and roll each into a ball. Do the same with your dough - divide it into 20 pieces and keep them covered while you work.
Take one piece of dough and roll it into a circle about 4 inches wide and as thin as a coin. Put a ball of filling in the center. Carefully bring the edges of the dough up and over the filling, pinching them together to seal completely. Make sure there are no holes or gaps.
Dust your mooncake mold with a little flour. Put the filled dough ball into the mold with the sealed side down, press firmly, then tap gently to release. Place on your lined baking sheet with the pattern facing up. Leave about 2 inches of space between each hopia.
Bake for 15 minutes, then take them out of the oven. Let them cool for 5 minutes, then brush the entire surface with your egg wash. Put them back in the oven for another 8-10 minutes until they're golden brown.
Let the hopia cool completely on the baking sheet for at least 1 hour before moving them. Store them in an airtight container at room temperature for up to 2 days, or in the refrigerator for up to 5 days. If you want to serve them warm, heat them in the oven at 300°F (150°C) for 5 minutes.