Set the sugar in a medium to large bowl. Zest the lemons over the sugar then rub it into the sugar to release the flavorful oils from the zest.
Melt the butter then add it, the oil, salt, extract, and vanilla and whisk very well.
Add the egg and whisk until the mixture is thick and shiny.
Add the flour, baking soda, and cream of tartar and stir to combine. Cover the bowl with plastic wrap and leave at room temperature for 2 hours.
Preheat the oven to 350 F. Grease then line two cookie sheets with parchment paper.
Portion the cookie dough into balls that are about 1.5-2 tablespoons (if you are topping with white chocolate, add the square of chocolate on top now). Bake until the edges are starting to turn golden, 10 minutes for smaller cookies, and around 13 for larger cookies.
Optional: out of the oven, drop the cookies once or twice on the counter (like pan-banging) to deflate. Alternatively, press them down with a measuring cup to flatten. If you prefer thick, domed cookies, leave them as is.
Let cool completely if planning to glaze. For the glaze: whisk together all the ingredients in a bowl. If the glaze is overly runny, add more powdered sugar. If it is too thick to drizzle, add more lemon juice.
For a zigzag glaze: dip the whisk in the bowl of glaze then hold it over the cookies and move it in a zigzag pattern. For full glaze coverage dollop glaze onto the cookie then spread. Zest some lemon over the top before the glaze sets, if you like. Let the glaze set on the cookies, then serve.