Beat butter, white sugar, and brown sugar with an electric mixer in a large bowl until smooth and creamy. Add eggs one at a time, mixing well after each addition. Mix in vanilla.
Combine flours, baking soda, salt, and cinnamon in a separate bowl; stir into creamed mixture. Fold in oats and dates. Cover and chill dough in the refrigerator for at least 1 hour.
Preheat the oven to 375 degrees F (190 degrees C). Grease 2 baking sheets.
Roll dough into walnut-sized balls and place 2 inches apart onto the prepared baking sheets. Flatten each cookie with a large fork.
Bake in the preheated oven until cookies are golden brown, 8 to 10 minutes. Cool cookies on the baking sheets for 5 minutes, then transfer to a wire rack to cool completely.