Combine dry ingredients in a large bowl: flour, baking soda, and salt. Set this bowl aside.
In another bowl (a stand mixer comes in handy here), beat eggs on medium speed with an electric mixer until frothy. Add the sugar and mix well.
In a cup or bowl, stir together the wet ingredients: milk, melted butter, vanilla, and butter flavoring. Add half of the dry ingredients and half of the wet ingredients to the egg/sugar mixture. Mix until combined. Add the the second half of the dry and wet ingredients, and also add in the sprinkles. Mix until fully incorporated!
Cover the bowl and let the dough sit in the refrigerator for about an hour. Letting the dough stiffen up will make it easier to cut out the doughnuts.
After an hour, fill a heavy-bottomed pot with about two inches of oil. Clip a candy thermometer to the pot and turn to medium low heat. While the oil is heating up, divide the dough into two or three sections. I find it a little easier to work with the dough in smaller portions. Plus, you can roll out one section and keep the other two in the fridge so they don't get too warm. Keeping the dough cold will make rolling it out/cutting out shapes much easier. Also when rolling out the dough, make sure you're working on a WELL FLOURED surface. Because this dough is soft and sticky, you need that flour to keep it from clinging to your work surface.
Once the oil has reached 360 degrees, gently drop one or two doughnuts into the oil. I usually do two at a time. The dough will look like it's sinking at first, but once it starts to cook a little, it will float to the top. Let the doughnuts cook for about a minute to a minute-and-a-half on each side. They should be a light golden brown color. (I cooked mine a little too long, so they're darker in the photos.) Use a slotted spoon to lift the doughnuts out of the oil and set them on a drying rack with a few layers of paper towels underneath.
(During this process, I would roll out more dough and cutting out doughnuts while the doughnuts were cooking, but you could also cut out all of your doughnuts beforehand and keep them in the fridge until ready to fry.)