Preheat oven to 350 degrees F (175 degrees C). Grease 12 muffin cups or line with paper muffin liners.
Beat brown sugar and butter together in a bowl; add egg and beat until creamy.
Whisk flour, cinnamon, baking powder, baking soda, salt, and allspice together in another bowl. Stir 1/2 the flour mixture into the butter mixture. Stir applesauce into butter mixture; mix well. Stir remaining 1/2 the flour mixture into butter mixture until just combined; add oats and chocolate chips.
Pour muffin batter into prepared muffin cups.
Bake in the preheated oven until golden and the tops spring back when lightly pressed, about 25 minutes.