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Cook raspberries, sugar, and ginger in a medium saucepan over medium heat, crushing with a potato masher, until softened and some of the liquid evaporates, 6–8 minutes.
Transfer 3 Tbsp. cooking liquid to a small bowl and whisk in cornstarch. Stir slurry into sauce in pan and cook, stirring, until thickened to about the consistency of jam, about 2 minutes; let cool.
Cook sugar and 3 Tbsp. water in a small saucepan over medium heat, without stirring, until mixture starts to darken in spots. Carefully swirl pan to evenly cook sugar but do not stir it. Cook, swirling occasionally, until caramel is a deep amber, about 8 minutes. Immediately stir in 1 cup cream. Cook until no streaks remain, about 30 seconds. Let cool.
Using an electric mixer on medium-high speed, beat remaining 2 cups cream in a large bowl to medium peaks. Add cooled caramel and beat to stiff peaks (a few swirls of caramel are okay).
Line an 8x8" baking dish with 2 sheets of parchment paper, leaving a 2" overhang on all 4 sides. Cover bottom with a single layer of cookies, breaking to fit as needed. Top with about ½ cup raspberry sauce and spread evenly to corners and edges. Top with ¾ cup raspberries, then spread 2 cups caramel cream over berries. Repeat layers with remaining ingredients. Cover and chill at least 6 hours.
To serve, top with more raspberries and currants (if desired) and slice, wiping knife between cuts.Do ahead: Icebox cake can be made 2 days ahead. Keep chilled.