Prepare a 9" pie pan by greasing with butter. Set aside.
In small saucepan, heat milk and butter over medium-low heat until butter is completely melted.
Remove from heat and set aside, cooling the mixture until it is warm to the touch and not too hot.
Fix your KitchenAid with the dough hook attachment and transfer the butter/milk mixture to the bowl of the KitchenAid. Add yeast, stir well, and allow to sit for 5 minutes.
Meanwhile, whisk together flour, cake mix, sugar and salt.
With mixer on medium-low speed, gradually add flour mixture and egg to butter mixture.
Increase speed to medium-high and stir for 5-7 minutes to knead. Dough should cling to itself and form a cohesive mass (if dough is too sticky, add additional flour, 1 Tbsp at a time, until it is manageable and does not stick to your hands).
Cover and place in a warm place for 15 minutes to rise.
Meanwhile, whisk together ¼ cup granulated sugar, ¼ cup brown sugar, cinnamon and sprinkles in small bowl.
Once dough has risen, transfer to a lightly floured surface and use a lightly floured rolling pin to roll out into an even rectangle about ¼" thick (about 14x9 rectangle)
Spread the ¼ cup butter evenly over the rectangle of dough (if it is too difficult to spread, microwave for about 5-10 seconds so that it spreads easily).
Evenly sprinkle the cinnamon-sugar mixture over the butter.
Starting from a longer end of the rectangle, tightly roll up the dough, pinching the end to make a seam.
Cut off about 1" of each end of the roll to create even cinnamon rolls and discard the ends. Cut the remaining roll into 12 even pieces and place into the prepared pie pan (the rolls should be touching one another).
Cover with a dish towel and allow to rise for 25-30 minutes. Meanwhile preheat your oven to 350F.
Bake 15-19 minutes (until edges begin to turn golden-brown).
While the rolls are baking, prepare your frosting by beating together butter and cream cheese.
Add powdered sugar and cake mix gradually until combined.
Stir in vanilla extract and then milk (start with 2 Tbsp, and then add another if it is still too thick).
Once rolls have finished baking, allow to cool 5 minutes before icing and sprinkling with additional sprinkles, if desired.