-HEAT TREAT FLOUR: See note 1; You can heat treat flour in the microwave or in the oven. Let flour cool COMPLETELY to room temperature before using. Don't use any burnt or clumpy flour (if it's off-color or smells burnt, it is burnt). The flour should be light, white, and fluffy. Use a spoon to scoop the cooled flour into a measuring cup and level the top of the measuring cup with the back of a table knife. Once measured (to 3/4 cup), set aside.
COOKIE DOUGH: Meanwhile, melt the 1/3 cup butter and set it aside to cool back to room temperature -- hot butter will melt the sugar and cause greasy or grainy cookie dough. In a medium-sized bowl, use a spatula to add every bit of the melted and cooled to room-temperature butter. Then add in the brown sugar, and white sugar. Briskly stir or whisk until smooth, about 1-2 minutes -- the mixture should be fully integrated and smooth (it changes consistency to be more creamy, but the butter shouldn't rise or separate). Once the mixture is smooth, add in the salt, vanilla, and milk or cream and stir until integrated. Stir in the fully cooled heat-treated flour and miniature chocolate chips. Mix until combined and smooth. Place the cookie dough in the fridge to chill while preparing the frosting.
FROSTING: In a large bowl, add 8 tablespoons of room temperature butter. Beat on medium speed for 3 minutes, or until completely smooth and fluffy, scraping down the sides of the bowls with a spatula as needed. Add in the vanilla, cream, and salt. Beat to combine. Finally, add in the powdered sugar and then beat the mixture (on low speed) to combine. Once combined, increase the speed of the mixer and beat until the frosting is very light and fluffy, about 3-4 minutes. (Frosting will turn a lighter white). Remove the cookie dough from the fridge and break/crumble it right on top of the frosting. Beat on low speed for about 10-20 seconds just to incorporate the cookie dough into the frosting; it's okay if there are rippled chunks of dough -- this makes the frosting have a fun texture!