Cut the butter into 12 small pieces, place on a plate and put in the freezer for 10 minutes.
In a small saucepan, combine the light and dark rums, lime juice, and shallot. Place over medium heat, and simmer until reduced to about 2 tablespoons of liquid, 6 to 8 minutes.
Remove the reduced liquid from the heat and whisk in two pieces of the chilled butter until smooth. Return to very low heat and continue whisking in the butter, 1 or 2 pieces at a time, until the sauce is creamy and emulsified, whisking all the time. Don’t let it get too hot, or the butter will melt and the sauce become oily; dodge the pan back and forth to keep the sauce just lukewarm enough to absorb the cold butter.
Remove from the heat and whisk in the mint. Set aside at the back of the stove, covered, while you cook the fish. (Don’t try to re-warm the sauce, or it may separate).
Preheat the oven to 400° and place a sheet of baking parchment on a rimmed cookie sheet. In a wide, shallow bowl, use a fork to thoroughly combine the salt, egg whites, and rosemary until the mixture is the consistency of snow. Place the steaks on the parchment and pat the salt mixture firmly into all four sides to create a fairly even coating that’s about ¼-inch thick.
Bake until the crust is hardened and slightly cracked, about 15 minutes. The internal temperature should reach 130°. Warm four dinner plates in the turned off oven with the door cracked open.
Crack the salt crust with the back of a knife and scrape away the crumbly crust. Lift each steak with a spatula, brushing away any excess salt from top and bottom with a dry pastry brush. Serve immediately on warm plates, spooning plenty of Mojito Butter on top.
Since you can’t see the fish, the best way to judge doneness is to insert a digital thermometer through a crack in the crust to the center of the fish.
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