Preheat oven to 350°F (175°C). Line baking sheets with parchment paper or silicone liners.
Whisk together the flour, baking powder, and salt. Stir in the oats. Set aside.
Using an electric mixer on medium speed, beat the butter, brown sugar, and peanut butter until thoroughly combined. Mix in the egg and vanilla.
Reduce mixer speed to low. Gradually add the flour mixture, mixing just until combined. Stir in the toffee bits.
Drop the dough by 2-tablespoon portions onto the prepared pans. Slightly flatten the tops of each cookie.*
Bake 10 to 12 minutes, or until the cookies are golden brown. Cool on the pan on a wire rack for 5 minutes. Then transfer the cookies to a wire rack to continue cooling.
Notes
*The cookies won't spread too much during baking, so be sure not to skip the flattening step.