In a large heatproof and freezer-proof bowl, whisk the agave nectar into the hot water until it has completely dissolved. Add the lemon and lime juices and whisk until incorporated. Freeze for at least 8 hours or overnight.
Slice a small portion off the bottom of each lemon and lime so that they can stand vertically on their own.
Slice a larger portion off the tops and hollow out with a melon baller until no flesh remains inside the fruit. Set aside.
Once the lemon-lime mixture is frozen, remove from the freezer and scrape it repeatedly with a spoon or the tines of a fork until the result is light, fluffy ice.
Spoon the icy lemon-lime mixture into the lemons and limes and garnish with mint leaves. Serve immediately, while they're chilly!