Mix flour, oats, baking soda, and salt in medium bowl.
Using an electric mixer, beat peanut butter, brown sugar, butter, honey, egg, and vanilla in large bowl until well blended. Stir dry ingredients into the peanut butter mixture in 2 additions. Stir in chopped chocolate.
Cover and refrigerate until dough is firm and no longer sticky, about 30 minutes.
Preheat oven to 350 degrees F (175 degrees C). Butter 2 heavy large baking sheets.
With hands, roll 1 heaping tablespoonful of dough for each cookie into 1 3/4 inch diameter ball. Arrange cookies on prepared baking sheets, spacing 2 1/2 inches apart.
Bake cookies until puffed, beginning to brown on top and still very soft to touch, about 12 minutes. Cool cookies on baking sheets for 5 minutes. Using metal spatula, transfer cookies to rack and cool completely. (Can be made 2 days ahead. Store in airtight container at room temperature.)