Preheat oven to 375 degrees. Line a baking sheet with parchment.
In a large bowl, whisk flour, sugar, baking powder, baking soda and salt. Place in bowl food processor and pulse until mixture resembles coarse meal. Return mixture to large mixing bowl.
Toss currants into the flour mixture, to coat. Separate any large clumps. Add half and half and gently stir until just comes together (mixture will be very dry). Add more cream by the tablespoonful if dough is too dry.
Turn dough onto lightly floured surface. Knead gently doing about three turns. Roll out dough to 1/2-inch thickness and cut withy 2-inch round cutter. (Do not twist when cutting). Gently fold scraps into each other, re-roll and cut.
Transfer scones to prepared baking sheet, spacing 2 inches apart. Brush with half and half and sprinkle with turbinado sugar. Bake until golden, about 20-22 minutes. Transfer to a rack to cool