Make the cranberry curd: To a medium, heavy-bottomed pot, add the cranberries, lime juice, a pinch of kosher salt, and 2 tablespoons brown sugar. Bring to a simmer over medium heat, cooking and stirring until the cranberries burst and soften, about 8 minutes. Remove from the heat and transfer the mixture to the bowl of a food processor, setting the pot aside. Let cool for 5 minutes, then process the cranberries, adding up to a tablespoon of water as needed to help the blade catch, until smooth.
Set a fine mesh strainer over a medium bowl and set aside. In a small bowl, whisk together the egg, egg yolk, and sugar, then beat in the cranberry mixture, then transfer back into the reserved pot. Place over medium-low heat and cook, stirring continuously and scraping the bottom of the pot with a silicone spatula, until the mixture has thickened enough to coat the spatula and reaches 170°F on an instant-read thermometer, 7–10 minutes. (Do not let it boil.) Remove from the heat and strain the curd, discarding any solids. Cool slightly, then whisk in the butter, a little at a time, until completely incorporated. Place a sheet of plastic wrap directly on the surface of the curd to prevent a skin from forming and refrigerate until completely chilled. (Cranberry curd will keep, well covered and refrigerated, for up to 5 days.)
Meanwhile, make the crust: Preheat the oven (with one of its racks positioned in the center) to 325°F. To the bowl of a food processor, add the hazelnuts, the remaining 2 tablespoons brown sugar, ginger, and a pinch each of salt and pepper and pulse until the nuts are finely chopped but not yet powdery. Transfer the mixture to a medium bowl, add the melted butter, then use a fork or your fingers to blend until the mixture is thoroughly moistened and holds together when squeezed in the palm of your hand. Lightly butter a 9-inch aluminum pie plate. Sprinkle the crumb mixture over the bottom of the pie plate, then use your fingertips to press it into an even layer, working the mixture into the corners of the plate and evenly up the sides. Cover the crust with a piece of plastic wrap and use the bottom of a measuring cup or glass to smooth and compress the crumbs into place. Transfer to the freezer until the dough is firm and set, at least 15 minutes or overnight.
Retrieve the pie shell from the freezer, place it on a large baking sheet, remove the plastic film, then immediately transfer to the oven and bake until just set and lightly browned, 8–10 minutes. Set aside to cool completely and turn the oven up to 350° F.
Make the lime custard: In a large bowl, whisk together the egg yolks and sweetened condensed milk. Add the lime juice, zest, and a pinch of salt, whisking until smooth. Line a large, rimmed baking sheet with parchment paper and place the cooled crust on it. Pour the filling into the shell, transfer to the oven, and bake for 15 minutes. Lower the temperature to 325°F and continue baking until the custard is firm around the edges and the middle still has a slight jiggle, 15–17 minutes more. Transfer the pie plate to a wire rack to cool to room temperature, then cover loosely with plastic wrap and refrigerate until completely chilled.
Finish the pie: Retrieve the pie from the fridge and remove and discard the plastic wrap. Use a paper towel to gently blot any condensation that has accumulated on the surface of the custard. In a large, chilled bowl, whisk together the mascarpone and powdered sugar. Use a silicone spatula to fold a few tablespoons of the heavy cream into the mixture to lighten, then add the remaining heavy cream and the vanilla. Use a handheld electric mixer or a large whisk to beat until thickened and smooth and soft peaks begin to form. (Take care not to overmix, which will cause the cream to become grainy.) Scoop about ½ of the mascarpone mixture onto the pie, then distribute ¼ cup of the cranberry curd in dollops over the cream. Use a fat skewer or chopstick to swirl the curd and cream together, then top with the remaining mascarpone cream and ¼ cup more of the curd and repeat. Cover loosely with plastic wrap or a large, overturned bowl and refrigerate for at least 15 minutes or up to 3 days. Cut into wedges and serve chilled, dolloped with the reserved cranberry curd, if desired.