In a small bowl, toss together sugar and lemon zest until well combined (you can use a mortar and pestle or rub using a fork).
In the bowl of a standing mixer fitted with the paddle attachment, combine flour, butter, salt, and baking soda and beat on low speed until it resembles cornmeal, 3-4 minutes. Add lemon sugar, cream, and vanilla and beat until just clumped together.
If the dough is soft, wrap it tightly in plastic and refrigerate for 30 minutes. Otherwise, on a lightly floured surface or parchment paper, work the dough until it comes together, then divide in half. Shape each half of dough into a log that is 1-1.5 inches in diameter. Wrap logs in plastic wrap and refrigerate or freeze (my preference) for at least 30 minutes or until set. Dough can be left in the refrigerator for up to 3 days or in the freezer for 2 months.
Preheat oven to 350F/180C. Line 2 pans with parchment paper.
Once the dough is cold enough, use a very sharp knife to slice the log into ½-inch thick rounds. They will look a bit thick, but they’ll spread during baking. If the dough is too crumbly and breaks, leave it for a few minutes to soften or just squeeze any broken bits back into each cookie.
Arrange cookies on prepared baking sheets, placing them 1-2 inches apart. Sprinkle lightly with granulated sugar and bake for 10-12 minutes or until golden brown. When done, the cookies should still appear quite soft, not firm. Sprinkle lightly with granulated sugar one more time if you like. Allow cookies to cool on a wire rack.
Serve cookies warm or at room temperature.
Store cookies at room temperature in an airtight container for up to a week or in the fridge for up to 2 weeks, bringing to room temperature before serving. Cookies can also be frozen for up to 2 months.