Make the pastry dough
1 Cut the butter into cubes. In a large bowl whisk together the flour, salt, and sugar. Sprinkle the butter cubes over the flour.
2 Use your clean hands to squish the flour and butter together, breaking up the butter into smaller pieces until the mixture resembles a rather shaggy coarse meal with a few pea-sized pieces of butter. Add the sour cream and a couple tablespoons of finely grated sharp cheddar cheese and stir with a fork until incorporated.
3 Gather the mixture with your hands and form a ball. Shape the ball into a disk. Sprinkle with a little flour, wrap in plastic wrap, and chill in the refrigerator for at least 30 minutes and up to a day ahead.
Prepare the galette
1 Peel, core, quarter the apples, place in water with lemon juice: Preheat oven to 400°F (205°C). Prepare a large bowl half filled with water. Squeeze into it the juice of half a lemon (and put the squeezed lemon in too if you want). Peel, quarter, and core the apples, and place into the acidulated water while you work to keep the apples from discoloring.
2 Slice the apple quarters: Remove the apple quarters from the water. Slice each apple quarter into 4 or 5 slices, lengthwise, about 1/4 inch to 3/8 inch thick at the widest end. Return the apple slices to the acidulated water.
3 Drain apple slices, sprinkle with maple syrup, cinnamon, minute tapioca, zest: Drain the apple slices. Place the apple slices back in the now empty bowl. Sprinkle with maple syrup and cinnamon and gently toss so that the apple slices are lightly coated with both. Add the minute tapioca and the lemon zest. Gently toss again.
4 Roll out the dough, transfer to lined baking sheet: On a clean, lightly floured, flat surface, place the disk of dough. Roll out with a rolling pin to an even thickness and about 14 inches wide. Gently move the rolled out pastry dough to a parchment lined baking sheet.
5 Arrange apple slices on the dough: Starting from 2 inches from the perimeter of the pastry round, arrange the apple slices, side by side, in a circular pattern, filling the circle. It's okay if some of the apple slices overlap a little.
6 Sprinkle with cheese, fold edges over in pleats: Sprinkle the top of the apples with grated cheddar cheese. Gently fold the dough up over the apple slices, forming pleats with the dough every few inches.
7 Brush with egg wash, sprinkle with coarse sugar: Place an egg in a small bowl with a teaspoon of water, beat with a fork or whisk until smooth. Use a pastry brush to brush the egg wash over the exposed border of the dough (don't put on the apples, just the dough). Sprinkle the egg washed crust with coarse sugar.
8 Bake: Bake for 10 to 15 minutes (until beginning to brown) at 400°F (205°C). Then lower the temperature to 375°F (190°C) and bake for 15 to 20 minutes more, until the apples are cooked through and the crust is nicely browned.
Cool until just lightly warm before serving.