Preheat the oven to 400F. Line a cookie sheet with parchment paper.
In a large bowl, cream together the butter and brown sugar. Add the eggs, one at a time, stirring well after each addition, then add the applesauce and stir until smooth. Add the flour, salt and spices, and mix just until combined.
Drop the dough by the tablespoonful onto the prepared cookie sheets, spacing the cookies about 2" apart.
Bake in preheated oven until set and lightly browned on the edges, about 9-12 minutes.
Let the cookies cool on baking sheets for 10 minutes, then transfer to a wire rack to cool off completely.
Make the Icing:
While the cookies are cooling, prepare the icing.
In a small saucepan set over medium-low heat, melt the butter. Continue cooking the butter until it's light brown and nutty-smelling. Immediately remove from the heat, and stir in the cream and vanilla extract.
Whisk in the icing sugar, ¼ cup at a time, until the icing is thick but still spreadable. Spread onto the cooled cookies, and let set completely before serving. Store any leftovers in an airtight container in the refrigerator.