Preheat the oven to 325 degrees F. Butter two 9-inch round cake pans, line the bottoms with parchment and butter the parchment. Lightly dust the pans with flour and tap out any excess.
Whisk the cake flour, baking powder and salt in a medium bowl. Beat the butter in a large bowl with a mixer on medium-high speed until creamy, about 1 minute. Add the brown sugar and 1/2 cup granulated sugar and beat until light and fluffy, about 5 minutes. Scrape down the sides and beat in the eggs one at a time. Beat in 1 tablespoon vanilla. Reduce the speed to low and beat in the flour mixture in 3 batches, alternating with the milk. Increase the speed to medium high and beat until the batter is just combined, about 20 seconds (do not overbeat).
Divide the batter between the prepared pans. Bake until the cake is just golden and springs back when gently pressed, 30 to 35 minutes. Transfer to a rack and let cool 20 minutes in the pans, then turn out onto the rack, remove the parchment and let cool completely.
Meanwhile, remove the ice cream from the freezer and let soften slightly. Combine both flavors in a large bowl and stir together with a rubber spatula. Line one of the cooled cake pans with plastic wrap and fill with the ice cream in an even layer. Cover with plastic wrap and freeze until firm but not rock solid, about 45 minutes.
Uncover the ice cream and invert onto one of the cake layers; discard the plastic wrap. Top with the second cake layer, gently pressing it into the ice cream. Smooth the sides with an offset spatula. Freeze until firm, at least 1 hour.
Meanwhile, pulse 2 cups strawberries in a food processor until liquefied. Strain through a fine-mesh sieve, pressing with a rubber spatula; set the puree aside. Combine the remaining 2 cups strawberries, remaining 2 tablespoons granulated sugar and the lemon juice in a small bowl; toss.
Combine the heavy cream, confectioners' sugar and remaining 2 teaspoons vanilla in a large bowl. Beat with a mixer on medium-high speed until stiff peaks form, about 3 minutes. Beat in the strawberry puree.
Remove the cake from the freezer and frost with the strawberry whipped cream. Return to the freezer until the whipped cream hardens, at least 45 minutes or overnight. Top with the macerated strawberries. Let the cake soften 10 to 15 minutes before slicing.