Preheat the oven to 400 degrees F.
Line a baking sheet with parchment paper.
Whisk 1 cup of the heavy cream, 2 tablespoons of the sour cream, and 1 teaspoon of the vanilla in a small bowl together. Refrigerate until ready to use.
Pulse 2 1/2 cups of the flour, 1/3 cup of the granulated sugar, the baking powder, and the salt in a food processor until combined.
Add the butter to the flour mixture and pulse until the mixture looks like a fine meal.
Pour the chilled heavy cream mixture into the flour mixture and pulse until just combined.
Using the extra flour, lightly dust a work surface.
Turn the dough onto the prepared work surface and pat it into a 1/2-inch-thick sheet.
Use a 2 1/2-inch biscuit cutter to cut out as many biscuits from the dough as possible.
Reshape the remaining dough into another 1/2-inch-thick sheet to cut out more biscuits; there should be about 12.
Place six of the dough rounds on the prepared baking sheet.
Brush the tops of the rounds with the extra heavy cream.
Top the cream-brushed rounds with the remaining biscuit round, making six dough sandwiches. This helps make the shortcakes easier to split later on.
Brush the tops of the biscuit stacks with the extra heavy cream and sprinkle them with the extra granulated sugar.
Bake the biscuits until golden and puffed, about 30 minutes.
Let the shortcakes cool for 5 minutes before transferring them to a wire rack to cool completely.
Toss the strawberries and 2 tablespoons of the brown sugar in a medium bowl.
Just before serving, beat 1 cup of the remaining heavy cream, the remaining sour cream, the remaining brown sugar, and the remaining vanilla in a large bowl with a mixer on medium-high speed until soft peaks form, about 2 minutes.
Split the shortcakes in half.
Spoon 1/2 of the cream mixture on the bottoms of the biscuits.
Top the cream layer evenly with the strawberries.
Top the strawberries with the remaining cream mixture.
Replace the biscuit tops and serve.