Preheat the oven to 350 degrees.
Mix together the starter, oil, flour, and sea salt.
Knead in the pesto until well-distributed.
Roll the dough a scant 1/4 inch thick on a baking stone or parchment-lined baking sheet. With a pizza cutter, cut the crackers into the desired shape. (I cut them into squares.)
Sprinkle with kosher salt.
Bake for about 20 minutes, until the crackers begin to turn brown. Turn off the oven, open the door a bit, and leave them in the oven for another 10 minutes so they become crispy.
Cool on a rack. Store at room temperature in an airtight container.