Preheat oven to 400 degrees F. Line a baking sheet with parchment paper.
In a large bowl, whisk the flour, sugar, baking powder, baking soda, and salt together.
Grate the butter over the flour mixture (it helps to use a frozen stick of butter). Toss to coat the butter in flour, then cut it in using a pastry cutter. A food processor could also be used. Add the chopped strawberries and toss to coat in the flour mixture.
In a separate bowl, beat the egg with the cream and vanilla extract. Add to the flour mixture, stirring with a fork just until a shaggy dough starts to come together. It's very much like a biscuit dough.
Turn the shaggy dough out onto the parchment-lined baking sheet. Use the paper to lift and press the dough together, kneading it a couple of times until it can be shaped into an 8-inch disc. If it's warm in your house, you might want to stick it in the freezer for a few minutes.
Cut the disc into 8 wedges using a sharp knife or bench scraper. Space them out on the baking sheet, then brush the top of each scone with a little extra cream and sprinkle with the coarse sugar. Refrigerate for 15 minutes so the butter is nice and cold before baking.
Bake for 22 to 25 minutes until the scones are golden brown on top.
While the scones are cooling, make the simple glaze by whisking the powdered sugar, cream, vanilla, and salt together until smooth. If you need to thin it out more, you can use additional cream.
Drizzle the glaze over the cooled scones. The glaze will dry and set after an hour, if you can wait that long.