Bring a large pot of water to a rolling boil, then score a small X on the bottom of each peach with a sharp knife. Drop the peaches into boiling water for 1 to 2 minutes, then transfer them immediately to an ice bath using a slotted spoon. Once the peaches cool enough to handle, peel away the skin, then pit and slice each peach into half-inch thick slices.

In a small bowl, whisk together the granulated sugar, brown sugar, nutmeg, cinnamon, ginger, salt, cornstarch, and tapioca starch. Combine these ingredients thoroughly so the starches distribute evenly and won't clump when they hit the liquid.

Place your sliced peaches in a large pot and pour the dry mixture over them, stirring gently with a wooden spoon to coat every slice. Let this sit for 5 minutes so the peaches begin releasing their juice and the sugar starts dissolving.

Pour in the water, lemon juice, and vanilla extract, stirring everything together gently until the mixture is combined. Make sure to scrape the bottom and sides of the pot to incorporate any sugar that settled there.

Place the pot over medium heat and bring the mixture to a gentle simmer, stirring occasionally with a wooden spoon. Continue cooking for 15 to 20 minutes, stirring every 2 to 3 minutes to prevent sticking on the bottom. The filling is ready when a spoon dragged through it leaves a brief trail before the liquid flows back together, or when the mixture reaches 220 degrees Fahrenheit on a thermometer.

Drop a small spoonful of hot filling onto a chilled plate and let it cool for 30 seconds, then nudge it with your finger to check the consistency. If it holds together but is still slightly wobbly, it's perfect; if it's too runny, simmer for another 3 to 5 minutes and test again.

Remove the pot from heat and let the filling cool to room temperature before transferring it to a pie shell or storing it. If you're making the filling ahead of time, spreading it on a baking sheet and refrigerating it will speed up cooling significantly.