Make the tart shells
Preheat oven to 400°F.
On a lightly floured surface, unfold, flour, and roll out the puff pastry to a square of at least 12 inches. Use a 5-inch round cutter or an overturned bowl and a paring knife to cut out four 5-inch circles.
Using a 3-inch round cutter, make a smaller circle inside each round. Press down so that the cutter goes most of the way through the layers of pastry, but not all the way through, leaving the centers attached.
Transfer the rounds to a parchment- or silpat-lined baking sheet.
Generously prick the inner circles with a fork to discourage rising.
Melt the butter in a small bowl. Stir in the honey until smooth.
Brush the honey butter over the surface of the shells.
Bake for 25 minutes, covering loosely with foil when the shells have browned to your liking. You can roast the grapes at the same time. Remove from oven and let cool. If centers have risen too much, you can gently press them down a bit to compress the layers and make more room for the filling.
Prepare the tart filling(s)
To make the sweet tart filling, place cream cheese, sour cream, and muscovado sugar into a mixing bowl or the bowl of a stand mixer fitted with the paddle. Beat until smooth, light, and creamy.
To make the savory tart filling, place goat cheese, lemon juice, salt, and pepper into a mixing bowl or the bowl of a stand mixer fitted with the paddle. You can use a fork to blend until smooth and creamy or beat with an electric mixer (or stand mixer paddle). If you use a fork and the mixture is a bit stiff, you can pop it into the microwave for a few seconds to warm it and make it perfectly spreadable.
Assemble the tarts
Spoon ¼ of the creamy filling into the center of each tart shell and spread evenly over the inner section.
For savory tarts, drape two pieces of prosciutto over the center of each tart.
Spoon some of the roasted grapes over each tart.
Drizzle with a bit of the syrup and serve warm or at room temperature.