Set aside 1/2 cup of raspberries. In a small saucepan over medium heat, combine the remaining 1 cup of raspberries with 2 tablespoons of sugar and the lemon juice. When the sugar has dissolved, turn the heat down and simmer, stirring occasionally, until the mixture starts to thicken, about 3 minutes. In a small bowl, whisk the water & cornstarch together. Stir this mixture into the saucepan and continue simmering until the mixture reaches a jam consistency, about 1-2 minutes. Stir in the reserved raspberries. Transfer the mixture to a bowl and set aside to cool to room temperature.
Preheat the oven to 350ºF (180ºC). Line two large cookie sheets with parchment paper.
In a medium-sized bowl, whisk together the all purpose flour, the cake flour, salt and baking soda until well blended.
In a large bowl, beat the butter, the remaining 3/4 cup of sugar, and the brown sugar, with an electric mixer at medium speed, until pale and fluffy. Beat in the egg and the egg yolk in 2 separate additions until well incorporated. Beat in the extracts.
On low speed, stir the flour mixture into the batter, just until combined. Stir in the white chocolate pieces. Fold in the cooled raspberry mixture very gently. Do not over mix! You want to see raspberry swirls. Over mixing will change the texture of the cookies.
Place 2-tablespoon sized portions of cookie dough about 3 inches apart onto the prepared cookie sheets. I used a #30 portion scoop.* Top each cookie with a piece of white chocolate and sprinkle with coarse sugar. Bake until the tops of the cookies are set and the edges turn light golden brown, about 10-14 minutes. Transfer to a cooling rack.