Step 1In a medium-size, wide heavy-bottom saucepan, combine the butter, water, salt, sugar and vanilla seeds and bring to a rolling boil over high heat. Remove from heat and quickly stir in the flour (stir quickly or the flour lumps will cook). Return the pan to low heat and cook, stirring frequently, for about 3 minutes to cook the flour slightly and rid the mixture of any starchy, floury taste.
Step 2Remove the pan from the heat and place the dough in the bowl of a food processor or stand mixer. If using a food processor, process the dough for 15 seconds to cool slightly and release steam (leave the tube open), then add all of the eggs and egg whites at once. Immediately continue to process for 30 seconds to combine and form the cruller batter. If using a stand mixer, beat the dough with the paddle attachment until most of the steam has subsided, then add the eggs and egg whites, 1 at a time, until each is incorporated and a batter is formed.
Step 3Remove the batter to a bowl set over an ice bath and continue to stir gently just until the batter cools and thickens (it should be thick enough to hold its shape when piped).
Step 4Grease a large piece of parchment paper with spray oil, or generously brush with oil to keep the doughnuts from sticking to the paper.
Step 5Place the batter in a large pastry bag fitted with a large star tip (the hole should measure just over one-half inch in diameter). Pipe the batter onto the greased paper into doughnuts measuring about 3 inches in diameter, with a hole about 1 inch in diameter, and spacing the doughnuts about 2 inches apart. The dough might slide as the doughnut is piped; hold the paper in place with one hand as you pipe, pressing in gently with the tip of the bag to close the doughnut and then quickly releasing to cut the dough (wet your fingers to pat down any dough "tips").
Step 6Fill a deep fryer with oil, or fill a large pot to a depth of at least 3 inches, and heat to a temperature of 350 degrees.
Step 7While the oil is heating, cut the paper around each of the doughnuts into squares. This will allow you to pick up each doughnut individually and invert it over the oil to release.
Step 8Gently place the doughnuts in the fryer one at a time, gently placing the paper in the hot oil until the doughnut releases and slides off. Fry just a few doughnuts at a time, being careful not to crowd. Fry the doughnuts until puffed and golden (they will triple in size), about 6 minutes total, flipping every 30 seconds or so. The doughnuts will begin to crack and expand after a minute or 2 -- this is fine, they will be slightly misshapen. Keep frying until evenly golden.
Step 9Drain the doughnuts on a rack and cool slightly, then frost and decorate as desired.