Preheat oven to 350F. Line two baking sheets with parchment paper or silicone mats.

In a large bowl, beat peanut butter, softened butter, brown sugar, and granulated sugar on medium speed until light and fluffy, about 2-3 minutes. Scrape down the bowl, then beat in the egg and vanilla until well combined.

Add flour, baking soda, and salt. Mix on low speed just until a soft dough forms. Do not overmix. The dough will be thick and slightly sticky.

Scoop out 2 tablespoon portions of dough and roll into balls. You should have 36 balls total. Flatten each ball into a thin disc about 3 inches wide - the dough will be fragile, so work gently.

Sandwich each Oreo between two dough discs, pressing the edges firmly to seal the Oreo completely inside. Roll gently between your palms to smooth into a thick ball. Roll each stuffed ball in the remaining 1/4 cup granulated sugar to coat.

Place cookies 2 inches apart on prepared sheets. Bake 11-13 minutes until the edges are set and tops are just barely crackled. Centers will look slightly underdone - this is perfect. Let cool on the pan for 10 minutes before transferring to a wire rack to cool completely.