Preheat the oven to 325 degrees F. Line an 8 inch square pan with nonstick foil or parchment.
In a food processor, combine the flour, sugar, oats, ginger, cream of tartar and baking soda. Pulse to combine. Add the butter and pulse on and off until mixture is crumbly.
Pat the (very crumbly!) dough into the prepared baking pan, pressing the surface with a piece of parchment (or a butter wrapper) until flat. Bake until the center is set and the top is pale, golden brown, 20 to 30 minutes.
Let cool to lukewarm in the pan on a wire rack. While still warm, carefully lift from pan and score into 16 squares. Do not separate. Let cool completely, then separate.