Add milk and butter to a small bowl. Microwave just until butter melts. The milk should be warm, but not hot. If it is hot, let it cool for a minute. Add the milk and butter mixture to the bowl of a stand mixer.
Add brown sugar, yeast, egg, and salt. Mix until just combined.
Add 1 3/4 cups of flour and mix with a dough hook for 1 minute. If the dough is still sticky, add a tablespoon of flour at a time until it is no longer sticky. Let the dough mix for 5 minutes.
Transfer the dough to a greased bowl. Cover and let it rise in a warm area for 60-90 minutes, until it has doubled in size.
Transfer the dough to a lightly floured surface. Roll the dough into a rectangle, about 12"x15". Spread the softened butter all over the dough, leaving a 1/2" border. Sprinkle on the brown sugar and cinnamon and use your fingers to evenly spread it across the dough.
Use a fork to mash the blackberries. Transfer the blackberries to the dough, trying to leave most of the juice behind (I like to use a fork or slotted spoon for this). Spread the blackberries evenly across the dough.
Fold the dough into thirds, folding the short ends in. Using a sharp knife or pizza cutter, cut the dough into 8 strips, about 1.5" wide.
Take each strip and use the knife or pizza cutter to cut the dough into three strips MAKING SURE to leave a 1/2" at the top that is NOT cut. Take the three strips and braid the dough, pinching it at the bottom.
Roll each strip into itself (like you're rolling a crescent roll), with the non-cut part in the center. Transfer the rolls to a greased muffin tin.
Preheat the oven to 375. Let the rolls rise for 30 minutes. Transfer to the oven and bake for 12-15 minutes, until rolls are golden brown.
Transfer the rolls to a cooling rack and making the icing. Combine powdered sugar and milk and whisk to combine. Start with 1 tbsp of milk and add more if the icing isn't runny enough. Drizzle the icing over each roll and enjoy while they're hot!