Preheat oven to 400°F and line a baking sheet with parchment or reusable liner.
In a large bowl, combine flour, granulated sugar, baking powder, salt, and baking soda.
Cut in butter using a pastry blender, two knives, or your fingers, until mixture resembles wet sand.
Stir in strawberries, making sure they get coated with flour mixture.
Stir in buttermilk, taking care to incorporate the dry ingredients well while not crushing the fruit. Mix until a shaggy dough forms,
Turn out dough onto a lightly floured surface and pat into a 1/2 inch thick rectangle. Fold once, pat again, and fold again. Repeat as needed for dough to look smooth but not be overworked, adding a small amount of flour as needed just to prevent sticking.
Pat into a rectangle or circle and cut out biscuits. Reshape as needed to cut remaining dough.
Place biscuits on a baking sheet, slightly separated. Bake for 18-20 minutes, until golden brown. Cool on a wire rack.
To make the icing, whisk together the powdered sugar and milk until smooth. Add more sugar or milk to get the desired consistency; it should be pourable but not too runny. Drizzle over cooled biscuits.