Peel, cube and cook the celery root in boiling salted water until fork tender. Peel, cube, and cook the potatoes in a separate pot of boiling salted water. For both of these, you want to start in cold water in the pot, bring it to boiling, then simmer until fork-tender.
When the celery root is tender, run it though a food mill or ricer on the finest setting. Taste the puree - either there will be small, tough fibers, or there won't. If there are tough fibers, run the puree through a fine-mesh sieve to remove the fibers. At this point the puree will be the consistency of applesauce.
Return the celery root puree to the pot and add the butter. Cook, stirring as needed, until the extra water is evaporated and the texture resembles mashed potatoes.
Meanwhile, rice the potatoes.
When the celery root puree has reached the correct consistency, add the riced potatoes to the pot, along with the heavy cream. Stir to combine, then taste. Add salt, if needed, and add more cream if the potatoes seem dry. Add several generous grinds of black pepper, stir to combine, and take the mixture off the heat. Let it cool a bit before adding the egg, or refrigerate and continue the next day.
When you're ready to continue, heat the oven to 375 degrees. If you've just taken a roast out of the oven, you're already there, or at least close. If you need to cook other things at other temperatures, it's fine. The potatoes won't mind if they're cooked at a slightly higher or lower temperature - they'll just take a little more or less time to cook.
Since the vegetables are completely cooked, we're just looking for some browning, some melted cheese, and a hot interior to cook the egg.
Line 2 baking sheets with silicone mats.
Add the eggs to the potatoes and mix well. Working in batches, put the potato mixture in a piping bag fitted with a large decorative tip. Pipe mounded circles of potatoes onto the baking sheets. I made 24 mounds, 12 on each sheet. Be as decorative as you like. You can make them all the same (hah! as if!) or make each one different (more likely!)
For an added garnish, grate cheese (I used a mild cheddar) on top of some or all, or put a small cube of cheese in the center. Sprinkle with paprika, if desired. An extra sprinkle of black pepper or even some scattered herbs would also be fine.
Bake at 375 degrees until the cheese (if used) has melted and there's some browning on the edges of the potatoes and on the ridges of the piping design. Timing will depend on how warm or cool the potatoes were when you started, as well as oven temp and the size of your potato mounds, but check at 15 minutes and figure it might take up to 25.
Carefully remove the mounds from the trays using a small spatula for serving.
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