Pulse oatmeal, flour, butter, and salt in a food processor until pea-size crumbles form. Add water; pulse until dough forms. Transfer dough to a lightly floured baking sheet. Press a sheet of parchment paper over dough and using a rolling pin, roll dough into a ½" thick square; cover with a kitchen towel and let sit 1 hour to dry slightly.
Heat oven to 250°F. Using a 3" round cutter, cut out cakes; gather and reuse scraps. Transfer cakes to a baking sheet; bake, flipping once, until golden and slightly crisp, 1 hour and 15 minutes. The more slowly it cooks, the better the flavor will be. Let cool slightly; serve with butter and jam, if you like. Store in an airtight container at room temperature for up to a week.