Combine the yeast, water, sugar, evaporated milk, salt, shortening, eggs, and vanilla in the bowl of a stand mixer. Pour in 2 cups of the flour and knead on a medium-low speed with a dough hook until well combined. Slowly add in the remaining flour until a smooth dough ball forms. Dough should be wet, but should not stick to your fingers or the sides of the bowl.
Transfer the dough to a lightly greased mixing bowl, cover with plastic wrap or a clean dish towel, and let rise 1 1/2 hours.
Roll the dough out onto a lightly floured surface 1/4 to 1/2-inch thick. Use a drinking glass or round cookie cutter to punch out 12 donuts, re-rolling the dough as necessary to get a dozen donuts and use almost all of the dough. Let rise another 30 minutes.
Meanwhile, pour vegetable oil into a large pot, enough for the oil to be 1-inch deep. Heat to 350 degrees Fahrenheit. While the frying oil is coming to temperature, make the glaze in a medium-size mixing bowl by whisking together melted butter, powdered sugar, pinch salt, 2 tsp evaporated milk, and vanilla extract. Add in water 1 teaspoon at a time until it is just thin enough to dip something in easily. It shouldn't be runny, but it shouldn't be thick either.
Fry the donuts a few at a time until golden brown, flipping halfway through. It is helpful to do a test donut to ensure that the donut gets cooked through. Transfer the donuts to a wire rack covered with a layer of paper towels. Let cool for 1 to 2 minutes then dip the top of the donut into the glaze and place it back on the wire rack.
Instead of frying in vegetable oil, you can also use vegetable shortening instead. You'll need approximately 2 cups of shortening.