1. Preheat the oven to 250 degrees F. Whisk 1 1/2 cups flour, baking powder, cinnamon and salt in a medium bowl.
2. In a large bowl, whisk the eggs, milk, sugar and vanilla. Melt 2 tablespoons butter in a medium nonstick or cast-iron skillet. Whisk into the egg mixture and reserve the skillet for cooking the pancakes. Whisk the dry ingredients into the liquid mixture until it forms a thick, lumpy batter.
3. Core and grate the apple on the large holes of a box grater into a small bowl. Toss with the remaining 1 tablespoon flour. Add the caramel sauce and stir until the flour is absorbed. Stir into the batter until just combined.
4. Melt 1 tablespoon butter in the reserved skillet over medium heat. Drop 2 tablespoons batter per pancake into the hot skillet. Cook until the surface of the pancakes form bubbles and the edges are set, about 3 minutes. Flip the pancakes and continue to cook until puffed in the center, about 1 minute more. Transfer to a baking sheet and place in the oven to keep warm while making the rest of the pancakes. Continue to make the pancakes, adding the remaining 1 tablespoon of butter as needed, until all the batter is gone.
5. Serve the pancakes with a dollop of yogurt, whipped cream, or, for dessert, a scoop of ice cream. Drizzle with additional caramel sauce.