2 pounds cream cheese, at room temperature for at least 6 hours
1/2 cup apricot jam
1 1/3 cup plus 1/4 cup granulated sugar
1/2 cup shelled, unsalted coarsely ground pistachios
3/4 teaspoon coarse salt, divided
3 1/2-pint containers raspberries
2 teaspoons pure vanilla extract
1/4 cup shelled, unsalted pistachios
1 tablespoon vanilla paste, or seeds from 1 whole vanilla bean
1 1/4 cup graham cracker crumbs
5 large eggs, at room temperature
5 tablespoons unsalted butter, melted and cooled
2/3 cup sour cream
Preheat the oven to 300°F. Bring water to a boil for a water bath. Butter the bottom and sides of a 9 by 2-inch cake pan; set aside.
For the filling: In the bowl of a standing mixer fitted with the paddle attachment, place the cream cheese, 1 1/3 cup sugar, 1/2 teaspoon salt, vanilla extract, and vanilla paste or seeds.
Beat on medium-high speed, scraping down the bowl several times, until the mixture is completely smooth, about 5 minutes.
Turn the mixer to low and beat in the eggs one at a time, until blended, scraping down the sides of the bowl after each addition. Beat in the sour cream until blended.
Place the cake pan in a roasting pan. Pour the cream cheese mixture into the cake pan—it will come all of the way to the top of the pan. Place the roasting pan in the oven and pour in the boiling water to come about 1/2 inch up the sides of the cake pan.
Bake until the filling is set but still jiggles slightly in the center, 1 hour 15 minutes to 1 hour 20 minutes. Remove from the water bath and let cool to room temperature. Chill for 4 to 6 hours, or overnight, in the pan.
To finish, heat the bottom of the pan over a flame or an electric element to loosen the crust. Invert the pan onto a 9-inch cake round or large plate.
Remove the pan from the cake and invert once more onto a 9-inch round or large plate so the crust is on the bottom.
In a small saucepan, warm the strained jam over low heat until liquid. Strain through a fine strainer. Brush the top and sides of the cheesecake with the strained jam.
Cover the top of the cake with a single layer of raspberries.
Holding the cake with one hand under the bottom, and working over a sheet pan to catch the excess, gently press the ground pistachios all around the side of the cake.
Dust with confectioners' sugar, if desired.