Preheat oven to 350ºF and line a few cookie sheets with parchment paper.
In the bowl of a stand mixer (or using a hand mixer), beat the peanut butter and butter together until smooth.
Add in the sugars and beat on medium-high until fluffy (3-4 minutes), scraping down the sides of the bowl as needed.
Add in the unsweetened applesauce and vanilla, beating on low speed until just combined.
Add in the flour, baking soda, and salt, and beat on low speed until the dough comes together. Dough will be slightly sticky, but moldable. If it’s still too wet, add up to another 1/2 cup of flour.
Using a tablespoon or cookie dough spoon, roll into 1″ balls and place a couple of inches apart on cookie sheets.
Press down the tops of each cookie lightly with a fork.
Bake for 10-12 minutes. Note that the cookies will seem slightly underdone, especially in the center. Don’t worry! They’ll finish baking once they’re removed and firm up as they cool. If you’re topping with flaky sea salt, sprinkle it on while they’re still very warm.
Let cool completely on baking sheets.
Store in an airtight container for up to a week!