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To make the dough: In a large bowl, combine the oil, butter, and sugars; beat until combined. Add the egg, extracts, and nutmeg, beating until smooth.
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Add the flour, baking powder, salt, and baking soda; mix until smooth. If piping the dough, skip the chilling step and transfer the dough to a large pastry bag fitted with an open star tip. If making drop cookies, cover the bowl and refrigerate the dough for at least 2 hours, until it's stiff enough to scoop easily and hold its shape; overnight is fine.
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To assemble and bake: Preheat the oven to 350°F. Lightly grease two baking sheets, or line with parchment.
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To make piped cookies: Pipe cookies onto the prepared baking sheet in desired shapes about 1 1/2" in diameter. If the dough feels (or becomes) too thick for you to pipe, remove it from the bag and add 1 tablespoon of water.
To make drop cookies: Alternatively, scoop the chilled dough by the tablespoonful (a tablespoon cookie scoop works well here) and roll in the sugar to coat. Place the balls of dough on the prepared baking sheets, leaving 1 1/2" clearance among them.
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Bake the cookies for 12 to 14 minutes, until the edges just begin to brown. Remove them from the oven and let them cool on the pan for 10 minutes before transferring them to a rack to cool completely.
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Dip the edges of the cooled sugar cookies in Fudge Frosting and then garnish with coarse sparkling sugar or sprinkles, if desired.
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Allow the cookies to set for 15 minutes before serving or transferring them to an airtight container to store at room temperature for up to 2 weeks. Freeze for longer storage.
