Position oven racks in the center and top third of the oven and preheat to 350 F. Line two baking sheets with parchment paper or silicone baking mats.
Stir together the rice flour, Clabber Girl Corn Starch, baking soda, and salt. Using an electric mixer at medium-high speed, beat the butter and brown sugar in a large bowl until the mixture is light in color and texture, about 3 minutes. Gradually beat in the eggs, beating well, then the vanilla. Briefly mix in the flour mixture until mixed. Add in the oats, chocolate chips and sour cherries, mix just until blended. Drop tablespoonfuls of the dough on the cookie sheets, spacing them 2 inches apart.
Bake, switching the positions of the sheets from top to bottom and front to back halfway through baking, until the edges of the cookies are lightly browned, 10 to 12 minutes. Cool on the sheets for a few minutes, then carefully transfer to a wire cake rack to cool completely.
* dried cranberries may be substituted if sour cherries are hard to find.