Cut parchment paper into about sixteen 4x4-inch squares, set aside.
Make the crullers: Place the milk, water, butter, sugar, and salt in a medium saucepan and bring to a boil over medium-high heat. Once boiling, remove the pan from the heat and add the flour, stirring with a wooden spoon until the flour is completely incorporated.
Place the pan back over medium heat and continue to cook and stir for 1 to 2 minutes to steam away as much water as possible. (The more moisture you can remove, the more eggs you can add later and the lighter your pastry will be.) The mixture is ready when a thin film coats the bottom of the pan.
Move the mixture to the bowl of a stand mixer fitted with the paddle attachment. Mix for about 1 to 2 minutes to allow it to cool. Then mix on medium speed and add the first egg. Let it mix in completely and then scrape down the sides of the bowl.
Add the remaining eggs, one at a time, and mix in completely until the paste becomes smooth and glossy. If the consistency is right, when you pull the paddle up the dough sticking to the paddle should hang down in a V shape ribbon that slowly breaks off. If your batter isn't the right consistency, add some of the additional whisked egg. Once your dough is the right consistency, transfer the batter to a piping bag fitted with a large open star tip.
Pipe circles onto the parchment paper squares, ending each with a little flick of your wrist and place on a baking sheet.
Fry the crullers: Heat at least 2 inches of oil in a heavy-bottomed pot until a thermometer registers 360-370°F (180-190°C). Working in batches, place two crullers at a time into the oil paper side up. Remove the parchment paper with tongs. Fry on each side until golden brown, about 3 to 4 minutes on each side. Remove fried crullers with tongs and drain on a paper towel. Repeat with the rest of the piped crullers.
Make the vanilla bean honey glaze: Whisk the powdered sugar, honey, milk, and vanilla bean paste together in a medium bowl until smooth and combined. Add more powdered sugar if the glaze is too thin or more milk if the glaze is too thick.
Dip cooled crullers in glaze and set on a wire rack until the glaze has set.