Beat butter in a large bowl with an electric mixer until creamy. Add 1/4 cup powdered sugar and 1/2 teaspoon vanilla extract; mix to combine. Add egg and mix until incorporated. Gradually beat in flour, thyme, and salt. Dough should come together and pull away from the sides of the bowl. Knead dough slightly and shape into a ball. Wrap in plastic wrap and refrigerate for 30 minutes.

Preheat the oven to 375 degrees F (190 degrees C). Grease a 9-inch round or 13x4-inch rectangular tart pan with a removable bottom.

Press dough evenly into the prepared tart pan.

Bake in the preheated oven until lightly browned, 12 to 14 minutes. Remove from oven and allow to cool completely before filling, about 30 minutes.

Combine 3 ounces of blackberries and 1/2 tablespoon sugar in a small saucepan over medium-low heat. Cook, stirring frequently and crushing the berries with a spoon, 4 to 5 minutes. Remove from heat and press mixture through a fine mesh sieve to remove all seeds and collect as much of the berry pulp as you can. Discard any seeds and solids left behind in the sieve and return crushed berries to the saucepan.

Mix together cornstarch and cold water in a small bowl. Add cornstarch mixture to the saucepan with the berry sauce; cook over medium heat until mixture comes to a boil and starts to thicken, 3 to 5 minutes. Remove from heat and let cool.

Combine mascarpone cheese, 3/4 cup powdered sugar, and 1 teaspoon vanilla extract in a bowl. Beat with an electric mixer until smooth.

Beat cream in a separate chilled glass or metal bowl with an electric mixer until frothy. Add sugar gradually, continuing to beat until stiff peaks form. Fold whipped cream into the mascarpone mixture and mix well. Spoon filling into the cooled tart crust and spread evenly. Using a teaspoon, drop spoonfuls of the smooth blackberry sauce on top of the filling. Using a knife, gently cut through sauce and filling to swirl. Garnish the top of the tart with the remaining 3 ounces blackberries and fresh mint.