To make the crust: In a small bowl, add the cold water and sourdough discard and stir with a fork to dissolve the discard into the water. Set aside.
To make the dough: In the bowl of your stand mixer fitted with the paddle attachment, add the flour and cold cubed butter.
Turn the mixer on low and slowly start to incorporate the ingredients. Gradually turn the mixer to medium speed. Once the butter mixture looks like wet sand, immediately add in all the water and sourdough mixture.
As soon as the dough comes together, stop the mixer. Do not over mix.
Turn the dough on a large piece of plastic wrap. Gather the dough together using the edges of the plastic wrap, this prevents your dough from heating up and the butter from melting. Flatten the dough into a disc and wrap it tightly in plastic. Refrigerate the dough for at least 4 hours, or overnight.
Preheat the oven to 400°F. Line a baking sheet with parchment paper.
To make the filling: In a large bowl, add the cane sugar, flour, cornstarch, and cinnamon and whisk together. Set aside.
Pit and slice the peaches and add to the sugar mixture.
To make the topping: In a small bowl, add the honey and water and microwave for 10 seconds to melt the honey. Set aside.
Remove the pie dough from the fridge. Remove the plastic wrap and place on a floured surface. Roll out the into a square that is 1/4-inch thick, it should be about 12 inches diameter.
Arrange the filling in the center of the chilled dough and spread it evenly, leaving a 2-inch border. Fold the edges of the dough up over the fruit.
Brush the dough with honey wash and sprinkle the turbinado sugar on top.
Bake for 35 to 40 minutes or until the crust has browned and the filling looks bubbly. Allow the galette to cool for at least 20 minutes before slicing and serving. (Be sure to check the recipe Notes for high altitude baking!)