Mix the all-purpose and rice flours together with the water in a large mixing bowl.
Gently tear or cut the chard leaves from the center ribs. Chop the ribs into ½-in (1.5-cm) pieces. Stack the leaves and roll lengthwise into cigars, then cut crosswise into ½-in (1.5-cm) ribbons. Keep ribs and leaves separate. You'll have about 1 cup ribs and 3 cups packed leaves.
Swirl 1 tablespoon of the oil in a medium nonstick or cast-iron skillet and set over medium-high heat until shimmering hot. Add the onion and chard ribs. Cook, stirring, for about 2 minutes, until the onions turn translucent. Add the ribs and continue to stir and cook for another 2 to 3 minutes. Stir in the leaves and a pinch of salt, then cook until they wilt, another 2 to 3 minutes. Remove from the heat and allow to cool for about 5 minutes. Fold the chard and onions into the batter.
Wipe out the skillet and heat 1 tablespoon oil over medium-high heat for 30 seconds. Pour half the batter into the skillet, tilting and swirling so the batter forms an even layer. The pancake should be about ¼-in (0.5-cm) thick. Cook, shaking the skillet occasionally, until the bottom is golden brown, 3 to 4 minutes. Lift up one edge and pour in 1 more tablespoon oil, then flip the pancake carefully and press down with a spatula to flatten it and ensure it cooks thoroughly.
Cook until the pancake is golden brown on the bottom and the edges are crisp, another 1 to 2 minutes. Turn and press down on the pancake 2 to 3 more times until the pancake is cooked through.
Transfer the pancake to a serving platter and keep warm in a 200°F (100°C) oven. Repeat with the remaining batter.
To serve, cut each pancake into 8 slices and serve with Soy and Green Onion Dipping Sauce.