Whisk together the water, yeast, milk, sugar, salt, and 1 1/2 cups of the flour until combined.
Add the egg replacer and beat until incorporated. Add the other 1 1/2 cup of flour, and using the dough hook, beat for 3-5 minutes until dough completely pulls away from the insides of the bowl. Once the dough is done, cover the top of the bowl with plastic wrap and let rise in a warm place until doubled in size, about one hour.
After the dough has risen, roll it out onto a floured surface at a thickness of about 3/4 inch thick and cut out using a small round cookie cutter. After cutting, place them on a greased pan, cover with plastic wrap and let rise for one more hour.
Put at least 2 inches of canola oil in a heavy pan and heat on medium/high for a few minutes until it reaches 375 degrees.
When the donut holes have risen to double their size, fry in hot oil for about 30 seconds per side. Check the bottom of the donuts and see if they are a nice golden color before flipping. Place holes on paper towels to drain excess oil and then dip them in glaze and put them on a cooling rack with wax paper under it to set.
Blend the following: 2 Cups powdered sugar (sifted if lumpy) 3-6 Tablespoons of water or non-dairy milk. Should be on the runny side.