For the beignets

1. Heat the water and milk to a warm bathwater temperature (not hot) in a large bowl. Add yeast and 1 tablespoon sugar and let sit for 10 minutes, or until the yeast is all frothy. 
2. Add the vanilla, melted butter, egg, and 3 more tablespoons sugar to the yeast mixture and whisk until homogenized. Keep whisking as you add the flour in three or four bouts. Keep mixing with a spoon until a shaggy dough forms. It will be something like a biscuit batter. 
3. Put the dough into a large greased bowl, and roll it around so that the whole thing gets coated in oil. Let the dough sit for three hours, or until it has tripled in size. You can also make the dough in advance and put it in a bag to refrigerate overnight. 
4. Punch down the dough and cut it in half. Place one half back in your greased bowl or bag and the other onto a well floured surface. Use your hands to pat the dough into a rectangle, then use a floured rolling pin to roll it 1/4 inch thick. Using a pizza slicer or large knife, cut the dough into 2-inch pieces.
5. Heat 2 inches of peanut oil in a small saucepan to medium. Fry each piece of dough for two minutes on each side, then an additional minute on the original side. After frying, remove the beignets and place them onto a paper towel to soak up any excess oil.
6. Dust with powdered sugar and serve immediately with hot coffee or tea and apple compote. 

For the apple compote

1. Peel the apples, slice them in half, and remove the seeds and stems.
2. Dice the apples like you would an onion, cutting lengthwise then widthwise. 
3. Add the apples, water, and sugar to a saucepan and bring them to medium heat. Turn the heat to low and cook covered for 20 minutes.
4. At the end of 20 minutes, add the cinnamon, nutmeg, and lemon juice, and cook covered for five more minutes or until all the apples have softened. 
5. Serve the apple compote warm with the beignets. Scoop the compote into the hole in the middle of the beignet to create your own filling.