Step 1Heat the oven to 400 degrees, and place a rack in the center of the oven. On a sheet of parchment paper, using a permanent marker, draw templates of the eclairs or cream puffs to help you pipe the batter. For the eclairs, draw 12 rectangles measuring 4 1/2 by 1 1/2 inches, spacing the rectangles about 2 inches apart (they will puff and spread as they bake). For the cream puffs, draw 2 1/2-inch circles, spacing the circles about 2 inches apart. Flip the parchment so the marker is on the underside of the sheet (you should still be able to see the templates), and place the parchment on a baking sheet.
Step 2In 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 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 3Remove 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 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 4Remove the batter to a bowl set over an ice bath and continue to stir gently just until the batter cools slightly and thickens (it should be thick enough to hold its shape when piped).
Step 5Dab a little of the batter underneath the 4 corners of the parchment paper so it sticks to the baking sheet. Place the rest of the batter in a large pastry bag fitted with a large round tip (the hole should measure just over one-half inch in diameter).
Step 6Pipe the batter evenly onto the paper to cover each of the templates; the piped batter will rise about one-half inch off the parchment paper.
Step 7Place the sheet in the oven and bake for 30 to 35 minutes, until the pastry is richly golden and evenly colored on the sides and top, and is firm when tapped. Turn off the oven and place the handle of a wooden spoon in the door to keep it barely open. Leave the pastries in the oven for an additional 30 minutes to give them time to dry out and set up.
Step 8Remove the pastries and place them on a rack, leaving a little space in between each. Prick the side or underside of each with the tip of a paring knife or skewer, and set aside until the pastries are cooled to room temperature, then use or fill as desired.