Position racks in the upper and lower thirds of the oven; preheat to 350 degrees F. Spread the oats on a baking sheet and bake on the top rack, stirring occasionally, until lightly browned and toasted, 8 to 10 minutes; let cool completely.
Whisk the flour, baking soda and salt in a small bowl. Beat the butter, brown sugar and granulated sugar in a large bowl with a mixer on medium-high speed until light and fluffy, about 3 minutes. Beat in the peanut butter until smooth, then beat in the eggs and vanilla. Reduce the mixer speed to low and beat in the flour mixture until incorporated. Stir in the oats and 1 1/2 cups chocolate chips with a rubber spatula until combined.
Line 2 baking sheets with parchment paper. Drop mounds of dough (about 2 tablespoons each) about 1 1/2 inches apart onto the prepared baking sheets. Flatten to about 1/2 inch thick with your fingers. Bake, switching the pans halfway through, until the edges of the cookies are set but the centers are still soft, about 18 minutes. Let cool 5 minutes on the baking sheets, then transfer to racks to cool completely.
Put the remaining 1/2 cup chocolate chips in a microwave-save bowl. Microwave in 30-second intervals, stirring, until smooth. Drizzle on the cookies. Let set, 15 minutes.