For the base and streusel topping: Preheat the oven to 350 degrees F. Butter the bottom of an 8-inch-square baking pan.
Put the flour, oats, sugars, baking soda and salt in the bowl of a food processor fitted with a metal blade. Pulse 6 times, about 3 seconds per pulse, until some of the oatmeal is finely chopped and some still large or whole. Drizzle in the melted butter, toss with a fork and then pulse until the mixture comes together, about 8 times.
Press two-thirds of the oat mixture into the prepared pan. Squeeze spoonfuls of the remaining mixture in your palm to make chunks - some larger, some smaller - and place in a small bowl.
Cover with plastic wrap. Refrigerate both the base and the chunks for 30 minutes.
For the filling: Put the apple butter, dried fruit, cinnamon and 2 tablespoons water in the cleaned bowl of the food processor and pulse to a chunky paste. Spread the filling over the chilled base. Scatter the streusel topping over the filling, leaving some of the filling showing through. Bake until the streusel is toasty and crisp, about 30 minutes. Cool completely on a rack, about 2 hours, before slicing into 16 squares.