Set aside about ¼ cup of the flour, and place the rest of the flour in the mixing bowl with the sugar, milk, yeast mix, egg, yolk and salt. Leave the excess flour (¼ cup) aside in case you need it later.
Mix on low speed with the paddle attachment or kneading hook in your mixer until the mix comes together.
If you used the paddle mix, switch to the dough hook and start kneading the dough.
Add the softened butter (1/2 - 1 tbsp at a time) until it's all incorporated. Add some flour, if you need to get the dough off the sides of the bowl. Add the rest of the flour if needed to form a soft dough ball.
Knead the dough for about 5 - 7 minutes until you have a soft, shiny, slightly tacky dough.
Knead the dough for a couple of minutes in your hand to have a smooth dough ball.
Place the ball of dough in a lightly oiled bowl (with enough room to rise) and cover with plastic wrap. Remember to place the dough, with the seam side down.
Let the dough rise at room temperature (70-75°F) till it is has doubled in size - this should take about an hour, maybe less depending on how warm it is.
Gently release the air in the dough and refrigerate it overnight at this point. This helps develop the flavours of the dough better and it's is easier to handle when it is chilled as well.
However you could roll out the dough and start cutting at this point if you wish. I recommend leaving it overnight (or at least a few hours) in the refrigerator.
Next morning, take the dough out on to a floured surface and roll out the dough till it's about ½ inch thick.
Cut out 3-3.5 inch diameter doughnuts with a doughnut cutter - or use a large and small biscuit or cookie cutter.
Place the doughnuts and doughnut holes on a parchment paper. Knead the left over doughnut dough lightly and let it rest for about 30 minutes, and repeat rolling and cutting once more.
Cover the cut doughnuts and doughnut holes with plastic wrap and let them rise for about 1 hour (see tips in the post to check if the doughnuts are proofed properly).
Heat a good amount of oil or shortening (the doughnut needs to float at least 2 inches above the bottom of the pan) in a heavy bottomed pan.
When the oil is heated to 375°F - carefully drop in a doughnut. After 30 - 45 seconds check to see if the doughnut has turned a golden brown. If it has, flip it over. If it hasn't leave it for a few seconds longer and then flip it over. The doughnut will darken as they cool, so be careful not to let it darken too much while frying.
Transfer the doughnut on to a wire rack to let it cool.
Dip one side in the pink glaze (or drizzle it over the top of the doughnuts), followed by rainbow sprinkles on top before the glaze sets.
For the Glaze
Sift the icing sugar and whisk in the milk or water, 1 tbsp at a time until you get a thick but pourable consistency. Add a couple of drops of pink colouring to get the desired colour.
Make more as needed.
Storage - doughnuts are best eaten on the same day or the next day, refrigerated. If you have too many doughnuts, freeze them without the glaze, then thaw them out later to eat with whatever glaze you like.
Recipe from The Flavor Bender by Dini Kodippili
The Simpsons Doughnuts https://www.theflavorbender.com/simpsons-doughnuts-best/