1. Set the oven rack in the middle position. Preheat the oven to 350 degrees. Line 2 baking sheets with parchment paper or spray with nonstick cooking spray; set aside.
2. Sprinkle the whiskey or rum over the raisins in a bowl and let stand while you assemble the batter.
3. Using a handheld or standing mixer, beat the butter, applesauce, brown sugar, granulated sugar and salt on medium-high speed until light and fluffy, 2 to 3 minutes, scraping down the sides of the bowl halfway through. Beat in the egg and egg white, milk and vanilla.
4. In a medium bowl, whisk together the flour, baking soda and cinnamon, add to the batter, and beat just until the flour is absorbed. Gently fold in the oats with a rubber scraper.
5. Use 2 spoons to scoop out the batter by rounded teaspoonfuls onto the prepared baking sheets, 1 1/2 inches apart. Bake until the cookies are set but still soft in the centers and beginning to brown on the edges, 17 to 20 minutes. (You can make larger cookies using a disher for desired size. My recipe actually only made about 20 cookies.).