Preheat the oven to 400 degrees F. Line 2 baking sheets with parchment paper
Combine the flour, sugar, baking powder, baking soda, and salt in a large bowl. Add the butter and cut in with a pastry cutter or a fork until coarse crumbs form.
Add the oatmeal and the blueberries and toss in until well blended.
In a medium bowl add the eggs, cream, and vanilla and beat until blended together. Gradually add to the flour mixture, tossing until the dough comes together.
Drop scones on the prepared baking sheet using a 1/2 cup measuring cup about 1 to 2 inches apart. Form the dough into mounds if needed.
Bake the scones for 15 to 17 minutes until the scones are golden brown. Remove from the oven and transfer the scones to a cooling rack to cool completely.
While the scones are cooling make the icing. In a small bowl add the powdered sugar and gradually add between 5 to 6 tablespoons of heavy cream or until the right consistency for drizzling the icing. Beat until smooth. Drizzle over the cooled scones.