Whisk the all-purpose flour, whole-wheat flour, cream of tartar, baking soda and salt in a medium bowl.
Whisk the eggs, buttermilk and sugar in a large bowl until foamy, then stir in the melted butter. Add the flour mixture and stir until just combined. (The batter will be thick; it's OK if there are a few lumps.)
Heat a large skillet over medium-low heat until hot. Lightly brush with vegetable oil. Drop a heaping 1/4 cupful of batter into the skillet for each pancake and spread into a 4-to-5-inch round with the back of a spoon. Cook until the bubbles on top burst and the edges begin browning, about 4 minutes. Flip and cook until golden brown, 2 to 3 more minutes. Repeat with the remaining batter, brushing the skillet with more vegetable oil between batches. Serve with butter and syrup.