For the pastry:
1 3/4 cups all-purpose flour
6 Tbsp. white cornmeal
2 tsp. sugar
3/4 tsp. salt
12 Tbsp. (1 1/2 sticks) cold unsalted butter, cut into 1/2-inch chunks
6 Tbsp. sour cream
1/2 cup ice-cold water
For the filling:
1/2 cup sugar
1/2 cup water
3 Tbsp. honey
3 Tbsp. fresh lemon juice
1/2 tsp. ground cinnamon
8 large Granny Smith apples, about 4 lbs. total, peeled, cored and sliced
1 1/2 cups fresh cranberries
2 Tbsp. unsalted butter, cut into thin slices
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.
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.
Position 2 racks in the middle of an oven and preheat to 400°F.
Divide the dough in half. On a lightly floured work surface, roll out each half into a round about 12 inches 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 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. Makes two 9-inch galettes; serves 10 to 12.