Preheat oven to 375°F. Line two cookie sheets with parchment paper.
In a mixer, combine butter, half and half, and vanilla on low speed until combined, then increase to medium speed and blend until smooth and fluffy, 30 -60 seconds.
Add powdered sugar and mix on low until combined.
Add flour and salt, then mix on low until large, moist crumbs form. Don't overmix.
Turn dough onto a flat surface. Using your hands, gently shape the dough into a flat disc. Wrap in plastic wrap and chill in the freezer for 5-6 minutes.
On lightly floured surface, roll dough until ¼" thick. Use rolling pin rings, if desired.
Use a 2¼ - 2½ inch circle cookie cutter to cut cookies. Gently gather the scraps and re-roll the dough until you have 24 cookies.
Place cookies on cookie sheet (12 per sheet) and freeze for 5-10 minutes or refrigerate for 15-20 minutes.
Bake chilled cookies for 10-12 minutes. Edges and bottom will be very lightly golden (tops will not be golden). **See picture in post.
Cool completely on cookie sheet, then move to cooling rack.