First, make the lime curd. In a large bowl, whisk together the granulated sugar, lime juice, salt, egg yolks, and half of the lime zest. Add the butter and set over a double boiler. Use a rubber spatula to stir the lime curd constantly until it starts to thicken. It should coat the spatula and cling to it before you remove it from the heat.
Strain the curd through a fine-mesh sieve into a medium bowl, then stir in the remaining lime zest. Cover the bowl and set it inside an ice bath to cool the lime curd until completely cold. The lime curd can be made ahead and refrigerated in an airtight container for up to 2 days.
When ready to make the ice cream, in a stand mixer fitted with the whisk attachment, whip the mascarpone, confectioners’ sugar, and vanilla until smooth, about 1 minute. Add the heavy whipping cream and whip the mixture until stiff peaks form, about 1 minute.
Mix one-third of the mascarpone mixture into the chilled lime curd. Fold half of the remaining mascarpone mixture into the lime curd until nearly blended, then add the remaining mascarpone mixture until completely incorporated. Pour into a 9 x 5-inch loaf pan and freeze until firm, at least 4 hours.