Preheat the oven to 400°F. Add the peaches, rum, sugars, cinnamon, vanilla, cornstarch & salt to a 10 inch skillet. Preferably cast iron. Stir to combine. Cover with foil and place in the oven for 10-15 minutes to soften and thicken a bit.
Add all the dry ingredients to a bowl and stir. Toss in the cold butter cubes and use your fingertips to squeeze the pieces into the flour until the mixture is coarse. Pour in the heavy cream and stir until a dough forms. It should look and feel like cookie dough. Set aside.
Mix the sugar, butter, egg and extracts until smooth. Add in the dry ingredients and mix into a paste. Set aside.
Lower the oven to 350°F. Remove the peaches from the oven and scoop the cobbler dough over the top of the peaches, alternating with the frangipane. Lightly brush the tops with heavy cream and sprinkle with coarse sugar. Bake for 35-40 minutes until the top is golden brown.
