Prep: Preheat your oven to 350° F. Use a paper towel to liberally grease a 10-cup bundt pan with shortening, making sure to coat all the nooks quite well. Add a handful of flour and tap it into all the nooks of the pan, coating the shortening. Tap out the excess. If you see any bare spots, be sure to go back and get them with shortening + flour. This is my no-fail method to prevent cakes sticking in the pan!
Make the Key Lime Cake batter: Beat the softened butter in a large bowl with an electric mixer on high speed for about 30 seconds, until creamed. Add the sugars and mix until completely creamed together and fluffy, about 1-2 minutes. Add the eggs, vanilla, lime zest, and lime juice and mix until just combined. Add the flour, baking powder, and salt. Mix on low speed just until the last streaks of flour disappear into the batter. Then, add the milk and greek yogurt and mix on low speed until combined. The batter will be relatively thick.
Bake: Bake for 55-60 minutes. A skewer or knife inserted into the pan (be sure to use something longer than a toothpick, since bundt pans are quite deep) should pull out some moist crumbs, but not wet batter. Let the cake cool in the pan on a cooling rack for 10 minutes. Then, loosen the edges with a butter knife and invert onto the rack to cool completely.
Make the Glaze: While the cake cools, whisk all the glaze ingredients together in a small bowl or glass measuring cup, starting with 2 cups powdered sugar. Taste, and add up to 1/4 cup more if it's a bit too tart for your taste.
Serve + Store: Pour the glaze over the cooled cake. Slice, and enjoy immediately! Store leftovers in an airtight container at room temp for 2-3 days.