For the crostata: Preheat the oven to 375 degrees F. Place the pie crust on a parchment-lined baking sheet.
Place a grill pan over medium-high heat or preheat a gas or charcoal grill. Drizzle the oil over the grill pan. Drizzle the peach halves with the balsamic vinegar, then grill flat-side down until softened and grill-marked, 3 to 4 minutes. Slice the peaches into wedges and arrange in a circle over the crust, leaving a 2-inch border. Fold the dough border over the filling to form an 8-inch round. Pleat the edge of the pastry and pinch to seal any cracks in the dough.
Combine the sugar and 2 tablespoons water in a small saucepan and cook without stirring until it has become a deep amber color, about 5 minutes. Remove from the heat and slowly whisk in the butter, lemon juice and salt. Immediately pour the caramel over the peach slices.
Bake the crostata until golden, 20 to 25 minutes.
For the thyme whipped cream: Meanwhile, combine the heavy cream, sugar, vanilla and salt in the bowl of an electric mixer. Whip until the cream has soft peaks, about 2 minutes. Gently fold in the thyme.
Let the crostata cool for 15 minutes to let the caramel settle, then cut into wedges and serve with a dollop of thyme whipped cream.