In a bowl, whisk together flour, cornmeal, salt and sugar. Using a pastry blender, 2 knives, or a food processor, cut in butter until it resembles coarse bread crumbs.
In a second bowl, whisk together sour cream and water. Stir into flour-butter mixture, about 2 tbsp at a time. If dough is still dry, add more water, 1 tbsp at a time, until dough holds together in one mass. Divide dough in half. Form each into a disc, wrap in plastic and chill at least 2 hours until firm.
Line a baking sheet with baking parchment. Roll out one piece of dough to an 11" circle and place on half the baking sheet (some of the dough will drape over the edge; this will be folded over the filling). Combine Monterey Jack with mozzarella and spread over center of dough circle, leaving a 2" rim. Sprinkle most of the basil over cheese, then top with tomato slices. Fold edges of dough up and over filling; there will be a 1-2" gap in the center. Sprinkle remaining basil into the gap.
Roll out second piece of dough to 11" circle and place on other half of baking sheet. Pour berries in center of dough, again leaving a 2" rim. Sprinkle with 1-2 tbsp sugar and dot with butter. Fold up edges of dough, again leaving a 1-2" gap. Brush exposed dough with water and sprinkle with 1 tsp sugar.
Bake galettes in preheated 400°F oven for 20 minutes or until golden brown. Cool about 10 minutes before serving.