Add flour, butter and salt to a food processor. Pulse on and off until the mixture is crumbly. Slowly add the water with the machine running just until the dough forms a ball. Chill for at least 2 hours or overnight.
Prepare the onions:
Heat olive oil on low in a large skillet. Add the onions with a pinch of salt. Sweat the onions, stirring, until they begin to caramelize. Turn up the heat to medium-low. Once a brown glaze has formed on the bottom of the skillet, pour approximately 2-3 tablespoons of water into the pan to deglaze and use a spatula to scrape up the brown bits and stir them into the onions. Repeat this glazing and deglazing process until the onions have been thoroughly caramelized and taste both sweet and bitter, approximately 25-35 minutes. Use more or less water as needed. Set the caramelized onions aside.
Prepare the galette:
Preheat the oven to 375 degrees F.
Roll the dough onto a lightly floured surface until very thin. Use a pizza cutter to make a circle with the dough that's around 12 inches in diameter. Carefully transfer the rolled dough to a baking sheet covered in parchment. The best way to do this is to gently roll the dough around the rolling pin, then unroll it on top of the baking sheet.
Distribute the shredded cheddar cheese evenly onto the dough, leaving approximately 2-inches around the border. Top with the caramelized onions.
Core the apple and slice it in half. Slice each half very thin, preferably using a mandoline on one of the thinnest settings. Fan the apple slices decoratively on top of the onions. Top with a light sprinkling of salt and pepper. Fold the excess dough on top of the ingredients in a circle, until you have a tightly formed tart.
Bake the galette until the dough is firm, approximately 18-22 minutes.
Notes
You can adjust the ratio of all purpose to whole wheat flour in the tart dough. The final photos in this post show dough with a higher percentage of whole wheat flour than the final recipe. I prepared the tart several times and was happiest with a ratio of 3 parts all purpose to 1 part whole wheat flour in terms of flavor and texture. That ratio will result in a dough that is lighter in color. The photos show the tart dough with a 50/50 ratio.