For crust: Mix flour and salt in processor. Add butter and cut in using on/off turns until mixture resembles coarse meal. Add 3 tablespoons ice water and process until moist clumps form, adding more water by teaspoonfuls if dough is dry. Gather dough into ball; flatten into disk. Wrap in plastic and chill 30 minutes. (Can be made 2 days ahead. Soften slightly at room temperature before rolling out.)
For filling: Melt butter in large nonstick skillet over medium heat. Add apples to skillet; sprinkle 3 tablespoons sugar over. Sauté until apples are golden and begin to soften, about 8 minutes. Add dried cherries and cinnamon and stir 30 seconds. Remove from heat and cool completely.
Preheat oven to 350°F. Roll out dough on floured surface to 12-inch round. Using 9-inch-diameter tart pan bottom as aid, transfer dough to rimless baking sheet. Arrange apple mixture atop dough, leaving 3-inch border. Fold edge of dough over apple mixture, pinching to seal any cracks in dough. Sprinkle remaining 1 tablespoon sugar over apple mixture and dough edge.
Bake galette 15 minutes. Increase oven temperature to 375°F. Continue to bake galette until crust is light golden around edges and apples are tender, about 35 minutes longer. Using tart pan bottom as aid, transfer galette to rack; cool 15 minutes. Serve warm with Crème Fraîche and Caramel Sauce.