Roast the squash until fork tender, I like to roast the squash face-down on a baking sheet at 425°F for 20-25 minutes. Move the squash to a cutting board, remove the skin, and mash it up well.
Make a small well in the middle of the squash and crack in the egg. With a fork, work concentrically from the outside in and whisk the egg with the butternut squash.
In two batches, add in the 350 grams of flour and continue to mix. Add in the salt, pepper and thyme leaves here as well. Mix until a very soft dough forms, shifting to using your hands when the dough comes together.
Move this dough to a floured surface and knead for 1-2 minutes, using the extra flour you have for dusting. Knead for just enough time to bring the dough into a smooth ball, but not too much that it becomes tough. You want this dough to be hard to work with, make it too firm and it will give you tough dumplings.
Dust off any excess flour and let the dough rest for about 20 minutes.
After the dough has rested, use a bench scraper to cut it into 8 equal pieces. Stretch those pieces out into cylinders and cut them in half again. These don't have to be perfect, but cut the dough into about 1" long pieces similar to gnocchi.
Use a floured finger to press down the center of each dumpling, creating little wells for the sauce to puddle into. Set aside and prepare the rest of the recipe.