1. Heat oven to 350 degrees F and line two baking sheets with parchment paper. In a large bowl, stir together flour, salt, baking powder, baking soda, and sugar. Use a pastry cutter to incorporate the butter until it's in pea-size pieces.
2. Stir in strawberries, then stir in heavy cream. Gently work the dough with your hands until it holds together (it will be slightly dry). If the dough is too dry, add more cream one Tbs. at a time.
3. Turn the dough out onto a floured surface. Dust your hands with flour and gently knead a few more times. Pat into a 1/2-inch-thick rectangle. Use a 1 1/2-inch round cookie cutter to cut the dough; space the rounds 2 inches apart on the baking sheet.
4. Place baking sheets in the refrigerator for 10 minutes. Bake until golden brown, turning halfway through, about 12 minutes. Let cool.
5. Stir together all the ingredients for the icing. Drizzle it over the scones, then allow the icing to set before serving.