Preheat the oven to 325 degrees F. Grease an 8-by-8-inch pan with butter. Cut a piece of parchment to hang over 2 sides of the pan.
Mix together the butter and almond butter by hand in a large mixing bowl. Beat the eggs in a small bowl, then add them to the butter mixture. Mix in the sugar, prune puree and vanilla.
Whisk together the cocoa powder, flour and salt in a separate bowl. Gently stir the flour mixture into the wet ingredients. Mix by hand just until combined; do not over-mix. Stir in the chocolate chips and walnuts. Pour the batter into the prepared pan. Sprinkle with the flaky sea salt.
Bake until a toothpick inserted in the middle of the brownies comes out clean, about 30 minutes. Remove to a wire rack to cool, and then cut into squares.