In the bowl of a stand mixer fitted with the paddle attachment, or using an electric hand mixer on medium speed, beat the softened butter. Add the sugar and mix until well combined.
Stir the egg, vanilla, lime zest, and lime juice and mix until just combined. Add the flour, cinnamon, and salt and mix just until the last streaks of flour disappear into the dough.
Divide the dough into two equal sections. Shape into logs that are about 8 to 9 inches (20 to 23 cm) long and 2 to 3 inches (5 to 8 cm) wide.
Pour your coarse or raw sugar onto a plate and roll each dough log through it to cover in the sugar.
Wrap each log separately in plastic wrap. Place each log inside a drinking glass laid on its side so it is encapsulated inside the glass. This helps keep the cookies round while they chill. Chill in the freezer for 1 hour or in the fridge for at least 2 hours, or overnight.
Preheat the oven to 350°F (180°C). Line two rimmed baking sheets with parchment paper or a silicone baking mat. Unwrap one of the dough logs. Move the other from the freezer to the fridge so it doesn’t freeze solid on you. Slice the log into 12 to 14 even slices.
Place the cookies at least 2 inches (5 cm) apart on one prepared baking sheet. If the cookie dough softened quite a bit while you worked with it, pop the sheet back in the fridge to chill for 5 minutes before baking. Repeat with the remaining dough log.
Bake one sheet of cookies at a time until the edges have a hint of golden brown, 15 to 16 minutes. Let the cookies cool for a few minutes on the baking sheet, then move them to a cooling rack to cool completely.