Place milk and yeast in the bowl of your stand mixer and allow to sit for 1 minute. 

Add the dough hook and mix in the flour, 1/3 cup of the sugar, salt, and eggs. Mix on low speed until dough starts to come together, about 5 minutes. 

Add the butter, a few pieces at a time, and mix until he dough becomes smooth and starts forms a ball, about 5 minutes longer.

Turn the dough out and form into a ball. Wrap tightly with plastic wrap and refrigerate over night.

Flour a large baking sheet; Set aside.

Turn out the dough onto a floured work surface and roll out to 1/2" thick. 

Using a 3 to 4 inch round dough cutter, cut out 9 rounds and place on the floured baking sheet. 

Cover tightly with greased plastic wrap and set side. Let sit in a warm dry spot for 2 to 3 hours, until doubled in size.

Heat oil in a pan with high sides or fryer to 350 degrees. Line a rack with paper towels. Place remaining sugar in a paper bag.

Working in batches, when the oil is hot, fry doughnuts, for 2 to 3 minutes a side, 2 or 3 at a time. Carefully flip, and continue to fry 2 to 3 minutes longer, until golden. 

Using a slotted spoon or wire basket, let oil drip from doughnuts, then transfer doughnuts carefully to sugar bag. Shake to cover doughnuts. 

Let cool on paper towel lined rack. Repeat until all doughnuts are fried.

Let cool completely.

Whisk the Irish cream, heavy cream, and sugar over high speed with whisk attachment in a clean mixing bowl until peaks form. 

Spoon into a piping bag fitted with a small star tip.

When doughnuts are cool, pop the piping bag filled with cream into the center of a doughnut, squeeze to fill until a bit pops out, making sure there is a good amount of filling in each doughnut. 

Repeat until all doughnuts are filled.