Combine brown sugar, butter, white sugar, and salt in a large bowl; beat with an electric mixer until a creamy, deep brown mixture forms. Add egg and vanilla; beat until mixture lightens and becomes smooth, 10 to 15 seconds.
Mix flour and baking soda together in a separate bowl. Add slowly to the wet ingredients until mostly incorporated, but some white traces of flour remain. Fold in milk chocolate, dark chocolate, and white chocolate using a spatula, not the mixer. Cover the dough and refrigerate for at least 30 minutes, or up to 48 hours.
Preheat the oven to 350 degrees F (175 degrees C) when ready to bake.
Divide cold dough into 2 1/4-ounce portions and place on a cookie sheet.
Bake in the preheated oven until the edges start to look golden brown and crispy, 10 to 12 minutes. Do not overbake; the centers will not look fully done. Cool until cookies are set, about 30 minutes.
For huge cookies, divide the dough into 5 portions and bake for 12 to 15 minutes.
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