Whisk the all-purpose flour, whole wheat flour, sugar, and salt in a large mixing bowl. Work in the cold, cubed butter with your fingertips until the mixture resembles coarse crumbs with some pea-sized pieces of butter remaining. Add the ice water and apple cider vinegar, then gently fold together until a dough forms. Shape into a 4-inch square, wrap tightly in plastic wrap, and refrigerate for 45 minutes.
On a lightly floured surface, roll the chilled dough into a, 8x11 inch square. Fold it into thirds like a business letter. Turn the dough 90 degrees, roll it out again, and repeat the letter fold. Complete three rounds of rolling and folding in total. Shape into a circle, wrap tightly in plastic wrap, and refrigerate for another 45 minutes.
Place the sliced baby bella mushrooms in a 10-inch skillet over medium heat without oil or butter. Cook, stirring occasionally, until they release their moisture and the liquid evaporates. Transfer to a large mixing bowl.
Add the olive oil, shallots, and thyme to the skillet. Cook until the shallots are tender and beginning to brown. Transfer to the bowl with the mushrooms. Add the sour cream, Dijon mustard, salt, and black pepper. Stir to combine, taste and adjust the seasoning, then allow the filling to cool.
Preheat a pizza stone in the oven to 400°F (200°C).
Place the chilled dough on a silicone baking mat and roll it into a 14-inch circle. Brush the dough with 1 teaspoon olive oil. Spread half of the mushroom mixture over the pastry, leaving a 2-inch border. Drizzle 1 teaspoon of olive oil over the filling and scatter half of the gorgonzola cheese on top. Add the remaining mushroom mixture and scatter the remaining gorgonzola over it.
Fold the pastry edges over the filling, overlapping the folds and leaving the center exposed. Whisk the egg with 1 teaspoon of water and brush over the folded edges. Reduce the oven temperature to 350°F (175°C). Transfer the galette from the silicone baking mat onto the preheated pizza stone. Bake for 40–45 minutes, until the crust is golden brown and the filling is heated through.
Transfer the galette to a wire rack and sprinkle with freshly chopped parsley, if desired. Allow it to cool for approximately 15 minutes before slicing.
Notes
Keep the butter and dough cold throughout the lamination process to maintain flaky pastry layers.
Cook the mushrooms until their released moisture evaporates to help prevent a soggy crust.
Prepare the dough up to 2 days ahead and refrigerate it tightly wrapped, or freeze it for up to one month.
Refrigerate leftover baked galette in an airtight container for 3–4 days, or freeze for up to 2 months.
Reheat leftover galette in a 350°F (175°C) oven for approximately 10–12 minutes to restore the pastry's crispness.