Beat brown sugar and butter together in a bowl using an electric mixer until smooth and creamy; beat in eggs and vanilla extract until smooth. Mix oats, flour, baking soda, and salt into creamed butter mixture until dough is well-mixed; fold in semisweet chocolate chips and white chocolate chips. Scoop 1/4 cup batter per cookie and place on a baking sheet 3 inches apart, partially flattening the dough.
Bake in the preheated oven until cookies are golden brown, 15 to 20 minutes. Cool cookies on baking sheet for 5 minutes before transferring to a wire rack.