To prepare the shortcakes, combine flour, sugar, baking powder, salt, and cardamom in a large deep bowl and stir to mix. Add the butter and cut it into the flour with a pastry blender or two knives until the mixture resembles coarse meal dotted with pea-sized chunks of butter. Add cream and stir to mix until the dough begins to clump together. Press the dough gently together into a mass without overworking it -- it should be crumbly and cracked.
Turn out the dough on a lightly floured surface and flatten it with your fingers into a 3/4-inch thickness (it's okay if it's rough and cracked so long as it mostly holds together). Cut as many 3-inch rounds as you can and transfer them to a baking sheet lined with parchment paper. Press scraps together gently and cut additional rounds, repeating until you've used all the dough. In an oven preheated to 375º F, bake until the bottoms begin to turn golden brown and the tops feel firm, about 15 minutes. Cool on a baking rack.
To prepare the filling, combine the peaches with granulated sugar, salt, and lime zest. Toss gently to mix and set aside for 10 to 15 minutes to allow the peaches to release their juices. You can refrigerate the fruit for several hours until ready to use.
Right before serving, combine heavy cream, powdered sugar, and vanilla extract in a separate bowl. Whisk with an electric mixer or by hand until soft peaks form.
After the shortcakes have cooled, slice each one horizontally through the center to make 12 halves. Working one at a time, place the bottom half of a shortcake cut side up on a plate. Add a large spoonful of peaches and their juices over the shortbread, followed by a generous dollop of whipped cream and the top half of the shortcake. Top with additional whipped cream, a thin slice of peach, and a light zesting of lime zest, if desired. Serve immediately.