BEAT TOGETHER THE BUTTER AND SUGAR with an electric mixer until just combined. Add the vanilla and mix briefly.
ADD THE FLOUR, SALT, AND LEMON ZEST on top of the butter mixture and mix on low until the dough begins to come together. If the dough is too dry and won’t hold together, mix in 1 or 2 teaspoons of lemon juice.
TURN THE DOUGH OUT onto a floured counter and pat into an oval. Wrap in plastic and refrigerate for 30 minutes. (If left longer, you may need to let it warm 10 or more minutes and/or beat it briefly with a rolling pin.)
PREHEAT THE OVEN TO 350 degrees F and set rack at center.
DIVIDE THE CHILLED DOUGH INTO THIRDS and pat each into a rectangle. Working with one portion at a time, roll the dough out to ¼-inch thickness and cut into cookies using a 2 ½-inch round, fluted cutter or another cutter of your choice.
PLACE COOKIES on a cookie sheet one inch apart, and sprinkle each with sugar. Bake one sheet at a time 10 minutes or until just golden on bottom and edges. Cool on racks. Store in a tightly covered container for a week or in the freezer for up to 3 months.