If your bread maker/machine has a "dough setting", use the bread machine method. If not, see stand mixer/manual instructions below in 'Notes'.
Method - bread machine:
Join the water, cream and vanilla extract, then add the melted butter. Pour into bread machine pan.
In a small bowl, add the sugar to the egg and beat well. Pour the egg and sugar into bread machine pan.
Add the flour into the pan, covering the wet ingredients.
Place the salt into the CORNERS of the bread machine pan. (So it doesn't immediately mix with the yeast.)
Lightly dig a shallow well in the middle of the flour, without reaching the wet ingredients. Pour the dry instant yeast in the well.
Set your machine to "dough" and let it do its thing.
When the dough has proofed (about 2 to 3 hours if you're not using a bread machine), turn dough out onto a lightly floured surface and roll out to about 1 cm (less than ½ incthickness.
Cut out 10 to 15 doughnuts (depending on the size of cutter you use).
Let proof for at least 30 minutes. Turn over and let the other side proof for another 30 minutes. The longer the proof, the better.
Frying:
If using a deep fryer, set to 170°Drop the doughnuts in, one or two at a time, top side down. (The last side to proof is the first to hit the oil.) Fry one side until golden, about 15 seconds, then turn over. Fry other side until golden, about 10 seconds. Remove from fryer and lay on paper towel.
If frying on a stove, measure the oil temp if you can or set to medium-high and keep a couple of pieces of dough left over from the doughnuts to test the oil. If the dough bubbles lightly in the oil and turns golden over 10 seconds, you're good to go. If there are too many bubbles and the dough browns within 10 seconds or less, your oil is too hot. If there aren't any bubbles, the oil isn't hot enough.