Make the dough: In a food processor, pulse to combine flour, salt and sugar.
Add butter and cream cheese and pulse until they form small pebbles.
Drizzle in lemon juice and 3 tablespoons ice water, then pulse just until dough comes together, adding another tablespoon of water if the dough looks dry.
Pat into a disc, wrap in plastic, and chill at least 1 hour and up to 3 days.
Make the strawberry compote: Combine approximately 2/3 of the strawberries in a small pot with the lemon juice and 1 to 2 tablespoons granulated sugar to taste, depending on sweetness of the berries.
Cook over medium heat until strawberries start to break down and juices are thick and syrupy, about 12 minutes. Let cool. Compote can be made up to 5 days ahead and stored in the refrigerator.
Make the frangipane: In a food processor, process 3/4 cups shelled raw pistachios and remaining 1/2 cup sugar until finely ground.
Add butter; pulse until combined, then add 2 eggs, the lemon zest and the almond extract and pulse until smooth.
Heat oven to 375 degrees.
Dust a work surface with flour.
Roll out dough to a 16-inch circle (it will be just slightly thicker than 1/8-inch), then transfer to parchment-lined baking sheet.
Leaving a 3-inch border around the edges, spread the frangipane filling evenly over dough.
Using a slotted spoon, dollop strawberry compote over frangipane, spreading any syrup gently so the compote and frangipane remain in separate layers, then fold dough edges up over the filling. About 6 to 8 inches of the filling should be exposed.
In a small bowl, whisk remaining egg with 1 tablespoon cold water, then brush over the dough.
Sprinkle with turbinado sugar and chopped pistachios.
Bake until edges are nice and golden, 40 to 50 minutes.
Transfer to a wire rack and cool completely.
To serve, top with remaining halved strawberries and dust with powdered sugar.