Place the sourdough starter in a medium bowl. Stir in salt, caraway seeds, onion powder, and fat of choice until thoroughly blended.
Add rye flour to the starter mixture. Use clean hands to mix it in well. Knead the dough briefly right in the bowl. Cover and let proof at room temperature about 8 hours or overnight.
Preheat oven to 350°F
Divide the dough into two portions. Roll out each piece of dough on a piece of parchment paper or a silicon baking mat. Roll dough as thin as possible.
Score the dough in 2-inch squares, using a rolling pizza cutter or a sharp knife. Prick each square with the tines of a fork to keep the dough from puffing up as it bakes. Sprinkle the dough with additional sea salt, pressing it lightly into the dough.
Slide the silicon mat or parchment paper onto a baking sheet. Bake 15 minutes, checking frequently after 10 minutes to make sure crackers are not getting too brown. Remove the outer rows of crackers before the center rows are fully baked, if necessary. Crackers should feel thoroughly dry to the touch. They will continue to crisp up as they cool.
Cool on a rack, separating any crackers that did not already break apart at the score lines.
Store in an airtight container at room temperature for a week or freeze for longer storage.