Preheat the oven to 325°F and grease a 9x13-inch pan.

Zest and juice the limes. Set the zest aside.

Stir the lime juice into the milk and let it sit while you prepare the batter.

In a small bowl, rub the lime zest into the sugar with your fingers until it smells fragrant.

In another bowl, whisk together the flour, baking powder, and salt.

In a stand mixer, beat the butter and lime sugar until light and fluffy.

Mix in the olive oil.

Add the eggs and egg whites one at a time, mixing well after each.

Mix in the vanilla and Greek yogurt.

Add the dry ingredients and lime milk in alternating additions, mixing just until combined.

Pour the batter into the pan and smooth the top.

Bake for 40–45 minutes, or until a toothpick comes out clean.

Let the cake cool completely before frosting.

Place the room temperature butter in the bowl of a stand mixer fitted with the paddle attachment and on medium-high speed mix for 4–6 minutes until the color changes from yellow to white.

While mixing, sift the powdered sugar using a fine-mesh sieve or sifter into a separate bowl.

Turn off the mixer and add the powdered sugar in 1–2 cup increments, mixing on low speed after each addition until fully incorporated. This prevents a sugar cloud.

Add the heavy cream, salt, and vanilla extract.

Mix on medium speed for 30 seconds to incorporate.

Increase to medium-high speed and mix for 5–10 minutes, pausing occasionally to scrape down the bowl.

Remove any air bubbles by pressing and smoothing the frosting with a spatula or wooden spoon in a zig-zag motion prior to frosting your cake.

Gently smooth the frosting on with an offset spatula. Garnish with freshly grated zest from 1 lime, a few lime slices for decoration, and a light sprinkle of salt on top.