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Preheat the oven to 375°F. Lightly grease (or line with parchment) two baking sheets, or a full-size or mini-scone pan.
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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.
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In a large bowl, whisk together the flour, salt, and baking powder.
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Work in the butter, using a mixer, your fingertips, a fork, or a pastry blender; the mixture should be unevenly crumbly.
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In a separate bowl, whisk together the strawberry/cream mixture, the egg, the vanilla extract, and the Fiori.
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Add this mixture to the dry ingredients, stirring until the dough just barely starts to come together.
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Add the 2/3 cup diced strawberries, gently folding them in; the dough will be quite sticky.
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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. Alternatively, divide the dough among the compartments of the scone pan of your choice.
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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). Remove the scones from the oven and break one scone open, just to be sure; they should be moist (but not gummy) at the center.
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To make the glaze: In a small bowl, stir together the sugar, water, and vanilla until smooth. Drizzle some atop each scone.
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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.