In a small, light colored saucepan, melt 1/2 cup (4 oz or 113g) of butter over medium heat, stirring occasionally, until dark golden and fragrant. You will see brown particles sink to the bottom. Keep a close eye on it. It can go from toasted to burned very quickly.
Remove from heat and immediately transfer the brown butter, including the brown particles, to a small bowl. Refrigerate until solid, about one hour.
Place the blackberries in a medium bowl. Sprinkle with granulated sugar and toss to coat. Let sit for 30 minutes until the berries release their juices.
Drain the blackberries and reserve the juice.
Preheat oven to 350ºF (180ºC). Line two large cookie sheets with parchment paper.
In a large bowl, whisk together the flour, baking soda, salt and cinnamon. Set aside.
In a large bowl, beat brown butter, softened butter and brown sugar until light and fluffy.
Add eggs and vanilla extract in three separate additions and beat until well combined.
Stir in the flour mixture just until combined.
Stir in the oats just until combined. Gently fold in the blackberries.
Scoop 1/4 cup sized portions of dough, about 2 inches apart, onto the prepared cookie sheets. *I used a #16 2 ounce scoop.
Bake until the cookie tops are set and the edges are golden brown, about 13-15 minutes. Transfer the cookies to a cooling rack to cool completely.
Pass reserved blackberry juice through a sieve to remove the seeds.
In a small bowl, whisk together the powdered sugar, heavy cream and one tablespoon of the reserved blackberry juice until smooth. Add more blackberry juice, a little at a time, until desired consistency is reached.