Cook squash until soft; puree until very smooth. Reserve 1 1/2 C squash puree in the food processor or blender; remove the rest for another use.
Add remaining ingredients, minus the fat. Process to blend. If time allows, refrigerate overnight. If not, try to allow a half hour or so for the liquid to absorb.
Heat generous quantity of fat until very hot. Form patties about 1 1/2 inches across. The squash mixture needs to be a bit dry looking in order to hold together. (The oil will resolve the dryness.)
Very gently turn the pancakes over after a couple of minutes. Try to let them brown a bit, but not burn. After another couple of minutes, carefully remove to a plate. They will be quite fragile, especially if you used a gelatin egg or the dough was a bit too moist. They still taste good even if they break!
If you don't use all the dough mixture up at first, it keeps very well in the refrigerator. In fact, I often prepare several days' worth at once and just use a bit each morning until we finish it all. Since there is no leavening, this will work.