In a large bowl, whisk together flours, baking powder, baking soda, sugar, and salt. Make a well in center; add eggs, butter, buttermilk, and vanilla. Whisk until just combined but still lumpy (do not overmix). Let stand 15 minutes.
Preheat a large cast-iron or nonstick skillet or griddle over medium. Drizzle with 1 teaspoon oil; wipe with a paper towel to leave a thin film. Pour 1/4 cup batter per pancake onto skillet, leaving 1 inch between each. Cook until bubbly on top and golden on bottom, and beginning to lift from skillet along bottom edges, 2 to 3 minutes. Flip and cook on other side until golden on bottom, puffed, and just cooked through, about 1 minute more. Transfer to a baking sheet lined with a wire rack; keep warm in oven on low heat until ready to serve. Serve with butter and syrup.