Preheat oven to 325 degrees F. Prepare an 8-inch square baking pan by lining the pan with aluminum foil, leaving 3-4 inches of foil draping over the sides of the pan.
Place graham crackers and gingersnaps into the bowl of a food processor and pulse until the crackers form fine crumbs, about 30 seconds.
To the crumbs, add butter, brown sugar, flour, and salt. Pulse until the ingredients are fully combined.
Using the bottom of a measuring cup or your fingers, press the crumbs into the bottom of the prepared pan.
Bake until the crust is beginning to brown, 12 to 15 minutes.
Prepare the filling while the crust bakes so that the warm filling can be poured into a still-warm crust.
In a medium saucepan, whisk together egg yolks, eggs, and sugar. Whisk in lime zest, lime juice, and salt. Add butter, and set over medium heat.
Cook the mixture, stirring constantly, until the mixture begins to thicken and it reaches 170 degrees F on an instant-read thermometer.
Immediately pour the mixture through a strainer set over a medium bowl and stir in heavy cream. Discard the lime pieces left in the strainer.
Increase oven temperature to 350 degrees F.
Pour the filling into the warm crust and bake until the filling is opaque and jiggles slightly in the center when shaken, 10 to 15 minutes. Be sure to rotate the pan halfway through baking.
Place the pan on a wire rack and let the bars cool for about 2 hours. Using the foil ends, lift the bars out of the pan, remove the foil, and cut into 16 squares. Dust with powdered (confectioner’s) sugar just before serving.