Preheat the oven to 375°F (190°C). Line a baking sheet with parchment paper or a silicone baking mat.
In a large bowl, whisk together 1 cup all-purpose flour, 1 teaspoon baking powder, ½ teaspoon baking soda, ½ teaspoon salt, and ¼ teaspoon allspice until evenly combined.
In a separate bowl, whisk together ½ cup granulated sugar, ½ cup packed light brown sugar, ¼ cup unsweetened applesauce, 2 tablespoons melted unsalted butter, 1 large egg, and 1 teaspoon vanilla extract until smooth and well blended.
Pour the dry ingredient mixture into the wet ingredients and stir until just combined; do not overmix.
Fold in 1 2/3 cups quick cooking oats until evenly distributed.
Add the 2 tablespoons miniature semisweet chocolate chips: either fold them into the dough now, or reserve some or all to sprinkle on top of each dropped cookie before baking.
Using rounded tablespoonfuls (about 1½ tablespoons per cookie), drop portions of dough onto the prepared baking sheet about 2 inches apart. This should make about 22 cookies.
If you reserved chocolate chips, press a few chips onto the top of each dough mound now so they show after baking.
Bake on the middle rack for 8 to 10 minutes, or until the cookies are set and lightly golden around the edges.
Remove the baking sheet from the oven and let the cookies cool on the sheet for 5 minutes, then transfer them to a wire rack to cool completely.