Prepare the peaches. In a big bowl, whisk together the lemon juice, 5 tablespoons of sugar, cornstarch, nutmeg and lemon juice. Slice the peaches into the bowl—I aimed for slices that were about 1/2 inch at their widest point. Discard the pits (by which I mean, add them to your compost pile.) Stir all of this gently just so everything is coated, and set the bowl aside.
Butter the baking pan.
Make the dough. In a medium bowl, add the flour, baking powder, baking soda, 5 tablespoons of sugar and salt. Stir with a fork. Then add the chopped butter. Work the butter into the flour with your fingers or a pastry fork or pastry cutter, crumbling everything together until the dough is the consistency of coarse cornmeal. Then add in the buttermilk and vanilla all at once and mix together quickly. Just stir until everything is incorporated together—don’t worry if there are a few small lumps.
Preheat the oven to 375ºF. Put the peaches in the baking pan and spread them around to make an even layer. Press down with your palm to even things out if need be.
Drop the dough on top of the peaches in six evenly spaced blobs. Don’t spread it around or stir it in or anything—just drop six big biscuity blobs right on top. They will spread somewhat as they bake.
Sprinkle the last tablespoon of sugar evenly on everything and put the pan in the oven. Set a timer for 45 minutes. This is ready when the peach juice is bubbling at the edges and the biscuit tops are golden—ours was ready at precisely 45 minutes.
Let it sit for half an hour or so to set up and cool enough that you don’t hurt yourself eating it. Serve by scooping sections into shallow bowls—we found that one of the biscuit sections was a nice post-dinner dessert for two (and one entire section would be a satisfying breakfast that will bring a smile to your face). Store leftovers in the fridge, covered well. If you want, reheat in a 325ºF oven for 10 minutes, or just let it sit out to warm to room temperature, or have it cold—a nice thing on a blistering day.