Place butter and sugar in the bowl of a large stand mixer, fitted with a paddle attachment OR in a large mixing bowl with a wooden spoon. Mix on medium speed or by hand until creamy, about 2 minutes. Add egg, vanilla, and lime zest beating well until smooth.
In a separate medium bowl, combine flour, baking soda, and salt. Fold into butter mixture until dough is just combined.
Scrape dough into a ball and wrap in plastic wrap. Refrigerate dough 2 hours or up to 4 days.
When ready to bake, preheat oven to 325F and line 2 large sheet pans with parchment paper.
Turn dough onto a floured surface and roll out to 1/4-inch thickness. Use a 4-inch round cookie cutter to cut out tea cakes and place 1-inch apart on prepared sheet pans.
Bake tea cakes at 325F 8-10 minutes until cookies are puffy and edges are a very light golden brown. Allow tea cakes to cool 10 minutes on sheet pan then transfer to a cooling rack to cool completely.