Peel and seed the squash. Peel the onion. Cut both into pieces that fit into your food processor chute.
Set up the food processor with the grater attachment. Grate the squash and onion.
Put the grated vegetables in a colander. Toss with half the salt. Let sit for 15-30 minutes.
Heat oven 250°. Prepare a baking sheet with a layer of paper towels.
Squeeze some liquid out of the vegetables using your hands or a rolled towel. Don't go too crazy with the squeezing, aim for about half the bulk you began with.
Remove the grater and put the blade in the food processor. Put half or more of the squash/onion mixture back in and process until smooth.
Combine the grated vegetables, processed vegetables, egg, flour, remaining salt, and caraway seeds in a mixing bowl. Stir to combine.
Heat the oil until a drop of water sizzles. Scoop using a heaping soup spoon, flattening slightly and reshaping in the pan. Do not overcrowd. You will fry the latkes in two or three batches.
Fry until brown. Flip and brown the other side. May be about five minutes total fry time, depending on the heat and type of oil.
Remove to the prepared sheet and keep warm in the oven while you fry the other batches.
Serve with applesauce and sour cream.