To make the pastry, in a food processor, combine the flour, cornmeal, sugar and salt. Scatter the chunks of butter over the top and pulse for a few seconds until the butter pieces are the size of small peas. In a small bowl, whisk together the sour cream and ice water. Drizzle the mixture over the dough and pulse until the dough is smooth and clings together. Pat the dough into a ball, wrap in plastic wrap and refrigerate for 20 minutes.

Position 2 racks in the middle of an oven and preheat to 400°F (200°C).

Meanwhile, to make the filling, in a large fry pan over medium heat, combine the sugar, water, honey, lemon juice and cinnamon and cook, stirring, until the sugar dissolves. Stir in the apple slices and simmer until the apples begin to soften, 5 to 7 minutes. Using a slotted spoon, transfer the apple slices to a bowl.

Add the cranberries to the liquid in the fry pan and simmer until they start to pop, about 2 minutes. Using a slotted spoon, transfer the cranberries to the bowl with the apples. Increase the heat to medium-high and boil the liquid until reduced slightly, then spoon over the fruit.

Divide the dough in half. On a lightly floured work surface, roll out each half into a round about 12 inches (30 cm.) in diameter and transfer to separate baking sheets. Divide the fruit filling equally between the pastry rounds and spread it in an even layer, leaving a 1 1/2-inch (4-cm.) border uncovered. Fold the border over the fruit, pleating the edges to form a rim. Lay the butter slices over the exposed fruit. Dust the pastry rims with sugar.

Bake the galettes, rotating them 180 degrees at the midway point, until the pastry is golden brown and the apples are tender, 35 to 40 minutes. Transfer the pans to wire racks and let the galettes cool completely on the pans.