In an electric stand mixer fitted with the paddle attachment, add the brown sugar, butter and granulated sugar and mix on medium until light and fluffy, about 4 minutes. With the mixer on low, add in the eggs one at a time, taking care to fully blend the first before adding the second. Be sure to scrape the sides and bottom of the bowl to make sure everything is incorporated. Add in the oil, milk and vanilla and mix 1 minute more. Take the bowl out of the mixer.
In a medium bowl, whisk together the bread flour, all-purpose flour, cream of tartar, baking soda and salt. Add this to the butter mixture and stir until the flour just disappears. Wrap tightly and put in the fridge for several hours, or until firm.
Preheat your oven to 350°F (177°C) and cover several baking sheets with parchment paper.