Cut parchment paper into 3 1/2 x 3 1/2 inch squares and coat the top of each with oil.
In a large saucepan, combine water, butter, and salt over high heat. Once the butter has melted and the mixture comes to a boil, remove from heat. Use a wooden spoon to immediately mix in the flour.
Place the pan back on the stove over medium heat. Continue to stir until dough forms and becomes a ball, about 1 minute. Transfer the dough to the bowl of a stand mixer or a large bowl and allow to cool.
Using a stand or hand mixer, beat the eggs, one at a time, into the cooled dough at high speed.
Fill a deep saucepan with 2-3 inches of oil and place over medium heat.
Attach a large star tip to large piping bag. Fill the bag with dough. Pipe rings of dough onto prepared parchment paper.
Once the oil is 350 degrees F, place the doughnuts, being careful not to overcrowd, into the pan paper side up. Gently remove the paper and fry the doughnuts on each side until golden. Use a slotted spoon to transfer to a towel lined plate. Repeat with remaining batter.
Place a wire rack over a sheet of parchment. In a large bowl , whisk lemon juice, rum, or water into powdered sugar until glaze comes together. If too thick, add more liquid. If too thin, whisk in more sugar. Dip each doughnut into the glaze and place on wire rack to set.