In a food processor, pulse flour, sugar, and a pinch of salt to combine. Add cold butter and pulse until mixture resembles coarse meal with some pea-sized pieces remaining, about 8 to 10 pulses.

Add egg yolk and drizzle ice water over mixture. Pulse just until dough starts to come together, 3 to 4 pulses. Turn out onto plastic wrap, press into a disk, and refrigerate at least 30 minutes or up to 2 days.

Heat oven to 375F. Divide dough into 6 portions. Roll each between parchment to 1/8-inch thick and press into tart pans. Trim excess and prick bottoms with a fork. Freeze 15 minutes, then bake until golden, 12 to 15 minutes. Cool completely.

Sprinkle gelatin over 2 tablespoons cold milk in a small bowl. Let stand 5 minutes to soften.

In a small saucepan, combine cream, remaining milk, and sugar. Heat over medium heat until steaming and sugar dissolves, about 4 minutes. Do not boil. Remove from heat and stir in vanilla.

Whisk softened gelatin into warm cream mixture until completely dissolved. Strain through fine-mesh sieve into a measuring cup. Pour into cooled tart shells, filling almost to the rim. Refrigerate until set, at least 3 hours or up to 24 hours.

Arrange fresh blackberries over set panna cotta. Brush lightly with warmed preserves for a glossy finish. Serve chilled within 2 hours of topping for best texture.