Preheat oven to 325°F (160°C). Line two baking sheets with parchment paper.
In a large mixing bowl, combine oil, applesauce, brown sugar, granulated sugar, and vanilla. Use a hand mixer on low speed to beat for 2 minutes until well combined and emulsified.
Add cocoa powder and beat until smooth.
Add flour, baking soda, and salt. Stir until fully incorporated. Dough should be thick but still soft.
In a small bowl, mix peanut butter, powdered sugar, and salt until smooth. Roll into 18 teaspoon-sized balls and set aside.
Place powdered sugar in a small bowl.
Scoop heaping tablespoons (~40 ml) of chocolate dough and roll into balls (18 total). Flatten slightly and make a small well in the center. Place one peanut butter ball inside, then wrap chocolate dough around it to completely enclose the filling. Roll into a smooth ball.
Roll each ball in powdered sugar to coat evenly. Place on prepared baking sheets, spacing cookies about 2 inches (5 cm) apart.
Bake for 12–15 minutes, or until edges are just firm.
Cool on baking sheets for 5 minutes, then transfer to a wire rack to cool completely.
Store in an airtight container at room temperature for up to 3 days. These cookies don’t freeze well as the powdered sugar will absorb moisture and become sticky.