Preheat the oven to 400°F. Line a baking sheet with parchment paper.
In a small bowl, combine the cinnamon and sugar. In another small bowl, melt butter in the microwave. Set both aside.
Peel, core and cut apples in half. Place one half, cut-side-down, on a cutting board. With a small, sharp knife, thinly slice the apple half into about 12 thin slices. Repeat with the remaining 5 halves.
Place the puff pastry on a cold surface (like a stone cutting board). Cut into 6 rectangles and place on the baking sheet about 1 inch or so apart. Layer the apple slices across the top of the puff pastry (see photos) until they nearly fill the length of the pastry rectangle. Leave a border of 1/4-inch all the way around. Repeat with the remaining rectangles of puff pastry and apple halves. Sprinkle each tartlet with the cinnamon/sugar mixture. Drizzle with melted butter.
Bake 25 minutes, until the pastry is golden around the edges and the apples are softened. Drizzle each with 1 teaspoon maple syrup. Serve warm or at room temperature.
*These are best when they are eaten the day they are made.