In a medium bowl, stir together flour, baking soda, and salt.
In a separate bowl, beat the butter and peanut butter for 30 seconds at medium speed, until creamy. Add the light and dark brown sugars, and continue beating 3 to 4 minutes, until the mixture is light in texture and color. One at a time, add the eggs, beating well after each addition. Beat in vanilla.
At low speed, a third at a time, beat in the flour mixture. Using a wooden spoon, stir in the semisweet and white chocolate.
Line three insulated cookie sheets with aluminum foil. Fill an ice cream scoop so that it is heaping slightly with cookie dough and drop on to one of the prepared cookie sheets. Continue, leaving 3 inches between cookies. Refrigerate for 30 to 45 minutes, until the cookie dough is chilled.
Position rack in the center of the oven and preheat to 350 F. Bake the chilled cookies for 17 to 22 minutes, until the edges are set and the centers are still soft. Do not overbake.