Position racks to divide the oven into thirds and preheat the oven to 350°F (175°C). Line two baking sheets with parchment paper or silicone mats.
In a medium bowl, whisk together the oats, all-purpose flour, baking soda, ground cinnamon, ground nutmeg, and salt until blended; set aside.
In a large bowl, beat the room-temperature butter, peanut butter, granulated sugar, and packed light brown sugar with an electric mixer on medium speed until smooth and creamy.
Add the eggs one at a time, beating for about 1 minute after each addition, then beat in the vanilla extract.
Reduce mixer speed to low and gradually add the dry oat mixture, mixing just until combined. Stir in the chopped bittersweet chocolate.
If desired, chill the dough for about 2 hours or up to 1 day for more evenly shaped cookies.
If dough is not chilled, drop rounded tablespoonfuls 2 inches apart onto the prepared sheets. If chilled, scoop rounded tablespoons, roll into balls, place 2 inches apart, and gently press each to about 1/2 inch thick.
Bake for 13 to 15 minutes, rotating sheets top to bottom and front to back after about 7 minutes, until cookies are golden and just firm around the edges.
Use a wide metal spatula to transfer cookies to cooling racks; repeat with remaining dough, cooling baking sheets between batches as needed.