1. Prepare the dough: Combine the eggs, oil, warm water, salt, and baking powder in a bowl, and beat with an electric mixer on medium speed until well combined. Reduce the speed to low and add 2 cups of the flour, 1 cup at a time, mixing well and scraping the bowl down after each addition. Using your hands, knead in the last cup of flour, a little at a time, until the dough is pliable and smooth. Wrap the dough in plastic wrap and refrigerate for at least 15 minutes or up to 8 hours.

2. Prepare the filling: Divide the oil between two large skillets and place them over medium-low heat. Add the onions to the two skillets and cook, stirring occasionally, until they are very soft and golden brown, 20 to 30 minutes.

3. Meanwhile, bring a large pot of water to a boil. Add the potatoes and simmer until tender, about 30 minutes. (Testing with a skewer, rather than a fork, will prevent them from breaking up.) Drain the potatoes, discarding the water, and return them to the pot over medium heat. Heat, stirring, for a minute or two to dry them out a bit. Then transfer the potatoes to a very large mixing bowl.

4. Add the sautéed onions (including the oil), egg whites, instant mashed potatoes, salt, and white pepper to the mixing bowl. Using a hand masher, mash the potato-onion mixture thoroughly. Set it aside to cool.

5. Cut the chilled dough into 6 equal parts. Roll out 1 piece of dough on a board to form a rectangle measuring about 8 × 14 inches (if it sticks, lightly flour the board; keep the remaining dough covered.) Place one sixth of the filling in a narrow strip along one long edge of the dough, about ½ inch in from the edge. Fold in ½ inch of both short sides, and roll up the log. The whole log (about 13 inches long) can then be stretched to the length of the baking pan you are going to use. Repeat with the remaining dough and filling. Place the logs on a baking sheet, cover with aluminum foil, and freeze (to facilitate cutting) until solid, 2 to 3 hours. When they are solidly frozen, wrap each log in plastic wrap and then in foil, and freeze.

6. About 2 hours before serving time, grease a baking sheet with shortening or cooking spray. Place as many of the frozen logs as you wish to use, seam side down, on the prepared baking sheet. (Each log will yield about 15 knishes.) Let them thaw until you can just get a knife through them, about 30 minutes. Then cut each log, with out cutting through the bottom layer of dough, into 1- to 1½-inch-wide slices, keeping the log together. (This will make it easier to cut them into individual knishes later while keeping the log together for baking.) Let the logs thaw completely.

7. Preheat the oven to 375°F.

8. Whisk the egg yolks and oil together in a bowl, and brush the tops of the knish logs with this mixture. Bake until golden brown, about 45 minutes. Cut into individual knishes, following the lines precut in Step 6. Serve hot.