Combine the cream with the pistachio paste, sugar, and salt in a saucepan. Bring to a boil while whisking occasionally. Pass through a fine-mesh strainer and cool over an ice bath.
Once the liquid has cooled, pass the egg yolks through a fine-mesh strainer and combine with the pistachio-cream mixture.
Preheat a convection oven to 120°C / 250°F with the fan on low speed.
Place a sheet of acetate on a cutting board and freeze.
Line a flat half sheet pan with plastic wrap and drizzle some corn syrup on it, then cut out an acetate sheet so that it fits inside the sheet pan. Press down so that the acetate adheres to the plastic wrap. Place this sheet pan inside a large sheet pan and place them in the oven; pour the pistachio brûlée base inside the smaller sheet pan, then pour hot water into the larger sheet pan to fill it three-quarters of the way. Gently push both sheet pans into the oven.
Bake until the custard has a “gelatinous jiggle” when the sheet pan is gently tapped. Slide both sheet pans out halfway. Lift the smaller sheet pan out of the larger sheet pan and let the custard set at room temperature.
Once the brûlée has cooled, freeze it.
Once frozen, flip the brûlée onto a cutting board lined with acetate. Cut the borders of the brûlée so that it fits inside the frame with the sponge cake. Do not assemble yet. Return to the freezer until needed.