In a large bowl and using a stand mixer fitted with a whisk or an electric hand mixer with a whisk attachment, whip together heavy cream, brown sugar, vanilla, and salt on medium-low speed until foamy, about 1 minute. Increase speed to high and whip until stiff peaks form, 1 to 3 minutes. Transfer to an airtight container and refrigerate if not used immediately.
Grilled Peaches:
Clean and oil cooking grate. Prepare a charcoal or gas grill fire for direct grilling over medium-high heat (450°F).
Wash and dry the peaches. Cut into halves, then remove the pit.
In a small bowl, stir together melted butter and vanilla.
Brush peaches with butter mixture. Dip the cut-side of each peach into the brown sugar mixture and rub the brown sugar into the butter, so it sticks. Place peaches cut-side down on grate directly over the heat and lightly press for better contact. Grill for 3 minutes. (If the fire gets too hot, move peaches to a cooler side of the grill until you see desired grill marks.)
Flip over and lightly sprinkle each peach with the remaining brown sugar mixture. Cook until sugar melts & peaches are soft, about 2 to 3 minutes.
Top each with brown sugar whipped cream and serve.