In a small bowl combine the warm water with a pinch of sugar, stir to combine. Pour in the yeast, stir and allow to sit and activate for at least 5 minutes. The yeast should bubble/foam. That's how you know it's working. If your yeast does not activate discard and start over with new yeast.
In the bowl of your stand mixer add the sugar, eggs, salt and vanilla extract. Stir to combine. In a small pot over low heat melt the butter with the milk. Remove from heat and allow to cool. Pour a few tablespoons of the warm milk mixture into the egg mixture and stir. This will temper the eggs. Now pour the rest of the milk mixture into the bowl. Add in the yeast water and 2 cups flour. With the dough hook attachment start your mixer. Once most of the dough has been incorporated stop the mixer and scrape down the sides of the bowl. Add the remaining 2 cups flour and mix to combine. The dough should form a ball but will be quite sticky. If the dough doesn't seem to be sticking together add in another 1/4 cup flour. Place in a slightly oiled bowl, cover and allow to rise in a warm spot for 1 hour. I often will turn my oven on low for a few minutes, turn it off and then let dough rise in the oven. I don't know about you, but my house is a bit chilly this time of year and yeast needs a warm environment in order to do their business.
Roll the dough out on a slightly floured surface. Use a 3-inch biscuit cutter to cut out donuts. Each should be 1/2-inch thick. I used the open side of a pastry tip for the holes, as I wanted extra small holes so the seeds would have more space to be once once they donuts were done. Place the cut donuts on a baking sheet covered with wax paper and a little flour, cover, and allow to rise for another 30 minutes.
After the second rise you can fill a large, heavy duty pot with 2-3 inches of oil. Heat to around 350°F. I like to use a candy thermometer to monitor my oil but it's not totally necessary. You can simply test oil with a scrap piece of dough before cooking.
Once the oil is ready place a few donuts in the oil. After about 30 seconds flip each donut. After flipping let them cook another 30-40 seconds. Then remove to a plate lined with paper towels to soak up any excess oil.
For the glaze whisk together 1/2 cup pomegranate water, 2 1/2 cups powdered sugar and 1/4 teaspoon vanilla extract. I added in some pomegranate seeds as well. Dip the warm donuts in the glaze. Then top with a few pomegranate seeds so they will stick to the surface.