Preheat oven to 425 degrees F. Place a sheet pan lined with aluminum foil on the bottom rack to catch drips.
First you need to peel the peaches. Fill a large stockpot with water and bring to a boil. Cut a small X into the bottom of each peach. Drop the peaches into the boiling water for approximately 1 minute. Using a slotted spoon, remove the peaches and immediately submerge into an ice bath to stop the cooking process. Gently peel the peaches using your hands or a paring knife (the riper the peach, the easier this will be). Cut the peaches into slices or chunks, discarding the pit. For the peach filling, stir together the sliced peaches and honey in a heavy pot or Dutch oven and allow to sit for a few minutes, until a syrup begins to form. (Be sure to adjust the amount of honey according to the sweetness of the peaches. If they aren't very sweet, add a little more and vice versa.) Add the cornstarch and bring the mixture to a boil over medium heat. Reduce the heat to low and simmer until thickened, about 7 to 10 minutes.
Remove from heat and whisk in butter and vanilla. Pour into desired baking dish (approximately 2 to 3 quarts) and set aside.
For the biscuit topping, whisk together flour, sugar, baking powder, and salt. Using your hands or a pastry blender, cut the shortening into the flour mixture until it is crumbly and well combined (it should resemble small pebbles). Stir in the cream and buttermilk until just blended.
Using your hands or an ice cream scoop, form round dumplings and drop onto the peach filling. Bake the cobbler uncovered for 25 to 30 minutes, until biscuits are light golden brown and the filling is bubbling. Carefully remove from the oven. Serve warm with vanilla ice cream.