Preheat the oven to 375°F. Lightly grease (or line with parchment) two baking sheets, or a full-size or mini-scone pan., To make the scones: In a blender or food processor, process 1/2 cup of the strawberries with the sugar and half-and-half or cream until the mixture is smooth. Set it aside., In a large bowl, whisk together the flour, salt, and baking powder., Work in the butter, using a mixer, your fingertips, a fork, or a pastry blender; the mixture should be unevenly crumbly., In a separate bowl, whisk together the strawberry/cream mixture, the egg, the vanilla extract, and the Fiori., Add this mixture to the dry ingredients, stirring until the dough just barely starts to come together., Add the 2/3 cup diced strawberries, gently folding them in; the dough will be quite sticky., Drop the dough by heaping tablespoonfuls onto the prepared baking sheets; each scone should be about the size of a golf ball. A tablespoon cookie scoop is exactly the right size for this task., Or divide the dough among the compartments of the scone pan of your choice., Make the glaze by stirring together the sugar, water, and vanilla until smooth. Drizzle some atop each scone., Bake the scones until they're just beginning to turn golden brown around the edges, 15 to 16 minutes (on a baking sheet or in a mini-scone pan), or 18 to 20 minutes (in a standard scone pan). Break one scone open, just to be sure; they should be moist (but not gummy) at the center., Serve scones warm. Or allow them to cool completely, and store airtight at room temperature. Just before serving, reheat very briefly in the microwave, or for 5 to 10 minutes, tented with aluminum foil, in a preheated 350°F oven.