In a medium saucepan, melt the butter over medium-low heat until melted. Remove from the heat.
Add the dark brown sugar and granulated sugar and stir until sugars are incorporated and smooth. Chill the mixture for 10 minutes. (Do not skip this step. You can also transfer the mixture to a different bowl so that it is no longer being heated from the pan.)
Remove the mixture from the refrigerator and stir in the eggs and vanilla.
Add the flour, oats, baking soda, salt, and cinnamon. Mix together with a spoon.
Gently fold in the white chocolate chips and semi-sweet chocolate chips.
Chill dough until it is firm enough to be hand-rolled, about 30 minutes in the refrigerator. Meanwhile, line cookie sheets with parchment paper.
Roll by hand into 24 medium-size balls, or use a scoop, and place on the lined cookie sheets.
Refrigerate for 30 minutes or freeze for 10 minutes.
While the dough is chilling, preheat the oven to 325°F.
Bake for 12 to 14 minutes, or until the edges just start to turn golden. (Do not overbake). Remove from oven and let cool a few minutes before transferring the cookies to a wire rack.