Combine butter and both sugars in a medium-sized bowl. Mix with a handheld mixer until smooth. All chunks of sugar and butter should be broken up.
Add flour and salt and mix just until combined. Mixture will be crumbly.
Add vanilla extract and milk. Mix until dough is soft.
In a food processor, take ¼ C baking chips and blend to break into tiny pieces. Add both chocolate baking chips and shredded baking chips last.
Roll dough into balls, about 1 tablespoon for each. Set aside.
In a medium sized pot, empty a full container of vegetable oil. Attach a candy thermometer to the right side of the pot. Heat oil over medium heat until it reach 360-375° F. Turn your heat down slightly to prevent the oil from getting too hot. Use extreme caution, oil will be very hot.
Take one of the biscuits and roll out very thin, to about the thickness of a nickel. Cut out circles about 2-2.5” in diameter. Wrap biscuit dough around a ball of cookie dough. Make sure to pinch the edges tightly and roll the biscuit dough in your hands to smooth.
Once oil is heated, slowly drop dough into oil. Bake for about 20-30 seconds a side until golden brown. Remove with a pair of tongs and set on a paper towel lined plate to drain the excess oil.
Allow to cool and roll in sugar.
Prepare the chocolate sauce. In a microwave proof bowl, combine 1 cup of chocolate baking chips and 4 tablespoons of heavy whipping cream. Microwave on high for 60 seconds. Remove from microwave and whisk until smooth.
When finished deep-frying, leave your candy thermometer in the oil until it is completely cooled.