In standing mixer, beat butter, sugar and molasses until light and fluffy, about 2 minutes, pausing to scrape down sides of mixer bowl occasionally. Beat in egg, then applesauce.
In a separate bowl, whisk together flour, baking soda, baking powder, salt and spices. With mixer speed on low, add to butter mixture and beat to combine. Chill dough while you preheat the oven to 375ºF (or up to 48 hours).
Place 1/3 cup sugar in a shallow bowl. Roll dough into 1-inch balls and then roll in the sugar. Bake 2" apart on parchment-lined baking sheets for about 8 minutes, until golden and crinkled.