Preheat the oven to 350°F. Set out several baking sheets and line them with parchment paper.: The moment you turn the oven on you will notice the small hum as it begins to warm, and by the time the dough is mixed the heat should be steady. Preheating ensures consistent oven spring and even baking, giving the cookies a uniform dome. A common mistake is skipping preheat, which causes uneven spread and underbaked centers, so always wait until the oven reaches temperature before you bake.
Set out an electric stand mixer. Combine the softened butter and sugar in the bowl. Beat the butter and sugar on high speed for 3 to 5 minutes, until light and fluffy. Scrape the bowl with a rubber spatula. Then add in the eggs, key lime zest, baking powder, vanilla extract, and salt. SAVE THE KEY LIME JUICE FOR THE GLAZE!: You will feel a change in texture as the butter and granulated sugar aerate; the bowl becomes lighter, and the mixture looks paler and somewhat mousse like. Proper creaming traps air which translates into a tender crumb, so be patient and scrape the sides to ensure even mixing. One pitfall is undercreaming, which yields denser cookies, so don't rush this stage.
Scrape the bowl again, then turn the mixer on low and slowly add in the flour. Once the flour is just combined, turn off the mixer as to not overmix the dough.: When you add all purpose flour , the dough will shift from glossy to more matte and start to come together in soft ribbons. Mixing on low prevents overdeveloping gluten, which keeps the cookies tender. If you overmix, the dough becomes tough and the cookies can be dense, so stop as soon as streaks of flour disappear.
Use a 1 1/2 teaspoon cookie scoop to portion out small balls of dough. Roll the balls in your hands and place them on the cookie sheets 2 inches apart. Bake the cookies for 10 minutes, until domed on top and barely golden around the edges.: As the cookies bake you will hear a very faint quieting as steam escapes; visually they will puff into domes and the edges will take on the slightest hint of gold. The centers should look set but not browned. A frequent error is leaving them too long; remove them when edges are just tinted to keep them soft inside.
For the glaze, set out a medium mixing bowl. Combine the powdered sugar, key lime juice, and heavy cream. Whisk well until very smooth. When the cookies come out of the oven allow them to cool for five minutes. Then dip the top of each cookie in the glaze, and flip them over onto a wire cooling rack.: The glaze should be glossy and pourable, coating the back of a spoon before you dip each cookie. After five minutes the cookies are cool enough that the glaze will set without soaking in, yet warm enough to encourage a thin spread. If the glaze is too thin the coating will run off, so add more powdered sugar to thicken; a common mistake is adding too much juice at once, making the glaze runny.
Allow the glaze to spread over the entire dome of the cookie. If desired, you can use extra key limes to add fresh lime zest across the top of the cookies while the glaze is still wet. Once the cookie glaze has set, you can stack the cookies in an airtight container for storage.: The glaze will smooth into a shiny cap and give a faint crackle when fully set; adding fresh key lime zest while the glaze is wet creates bright flecks of aroma and an attractive finish. Wait until the glaze is fully set before stacking to avoid sticking, and avoid packing cookies into a warm container which can make the glaze sweat and soften.