Make the purée: Add 1 1/2 cups raspberries with 2-3 tablespoons sugar and 1-2 teaspoons lemon juice to a small saucepan. Heat over medium until berries break down, about 3-5 minutes.

Mash until smooth and strain through a fine mesh sieve to remove seeds. Cool completely.

Preheat oven to 350°F. Line an 8x8 pan with parchment paper, leaving overhang.

Whisk 1 cup flour, 3/4 cup sugar, 1/2 teaspoon baking powder, 1/4 teaspoon baking soda, and 1/4 teaspoon salt in a bowl.

In another bowl, mix 1 egg, 1/3 cup oil, 1/3 cup milk, 2 tablespoons lemon juice, 1 tablespoon lemon zest, and 1 teaspoon vanilla.

Combine wet and dry ingredients until just mixed. Pour into prepared pan and bake 18-22 minutes until a toothpick comes out clean. Cool completely.

Beat 8 oz softened cream cheese with 1/2 cup powdered sugar until smooth. Mix in 1 teaspoon vanilla and 1/2 cup strained raspberry purée.

Fold in 1 cup whipped heavy cream. If using, fold in 1/4 cup melted white chocolate. Spread over cooled cake.

Refrigerate at least 3 hours or overnight. Before serving, drizzle with raspberry purée, add fresh raspberries and freeze-dried bits. Slice with a hot knife.