If you can't find Braeburn apples, Golden Delicious will work. Serve with vanilla ice cream or whipped cream.
1. For the topping: Adjust oven rack to middle position and heat oven to 400 degrees. Pulse flour, sugars, cinnamon, and butter in food processor until mixture has texture of coarse crumbs (some pea-sized pieces of butter will remain), about 12 one-second pulses. Transfer to medium bowl, stir in oats, and use fingers to pinch topping into peanut-sized clumps. Refrigerate while preparing filling.
2. For the filling: Bring fresh cranberries, 3/4 cup sugar, and water to simmer in Dutch oven over medium-high heat and cook until cranberries are completely softened and mixture is jam-like, about 10 minutes. Scrape mixture into bowl. Add apples, remaining 1/2 cup sugar, and dried cranberries to Dutch oven and cook over medium-high heat until apples begin to release their juices, about 5 minutes.
3. Off heat, stir cranberry mixture and tapioca into apple mixture. Pour into 13 by 9-inch baking dish set over rimmed baking sheet and smooth surface evenly with spatula.
4. To assemble: Scatter topping evenly over filling and bake until juices are bubbling and topping is deep golden brown, about 30 minutes. (If topping is browning too quickly, loosely cover with piece of aluminum foil.) Transfer to wire rack to cool. Serve warm.
MAKE AHEAD: While this crisp is best eaten the day it is cooked, the topping and filling can be made ahead. After pinching the topping into small clumps in step 1, transfer the mixture to a zipper-lock bag and refrigerate for up to 5 days or freeze for up to 1 month. The cooked filling can be refrigerated for up to 2 days. To bake, sprinkle the chilled topping evenly over the chilled filling, loosely cover with foil, and bake for 20 minutes. Uncover and bake until the juices are bubbling and the topping is deep golden brown, 15 to 20 minutes longer.