Step 1

Pulse 2 cups (250 g) all-purpose flour and 1 tsp. Diamond Crystal or ½ tsp. Morton kosher salt in a food processor to combine. Add ¾ cup (1½ sticks) chilled unsalted butter, cut into pieces; pulse until mixture resembles coarse meal with a few pea-size pieces of butter remaining.

Step 2

Transfer mixture to a large bowl and drizzle in 1 Tbsp. apple cider vinegar or white wine vinegar and ¼ cup ice water. Mix with a fork, adding more ice water by the teaspoonful as needed, just until a shaggy dough comes together (dough will look very dry).

Step 3

Turn out dough onto a surface; knead lightly to incorporate any dry bits of flour. Pat into a square and wrap in plastic. Chill at least 2 hours and up to 12 hours.

Step 4

Place a rack in middle of oven; preheat to 400°. Process 1 lb. crimini mushrooms in clean food processor until very finely chopped (just shy of a paste); transfer to a large bowl. Combine 3 large shallots, 1 garlic clove, and ½ cup coarsely chopped tender herbs (such as parsley and/or chives) in food processor and pulse until finely chopped. Scrape into bowl with criminis and mix to combine.

Step 5

Heat 2 Tbsp. unsalted butter and 2 Tbsp. extra-virgin olive oil in a large skillet over medium-high. Add crimini mixture (reserve bowl) and ½ tsp. Diamond Crystal or ¼ tsp. Morton kosher salt; season generously with freshly ground pepper. Cook, stirring occasionally, until browned and most of liquid has cooked off, 11–13 minutes. Scrape filling back into bowl; mix in 6 oz. sharp cheddar, coarsely grated (about 1½ cups), and ¼ cup heavy cream.

Step 6

Roll out dough on a lightly floured surface to a 16x10" rectangle. Transfer to a parchment-lined baking sheet. Spread filling over dough, leaving a 2" border. Scatter 6 oz. mixed mushrooms (such as oyster and/or king trumpet), torn, over. Fold borders up and over filling and brush dough with 1 large egg, beaten to blend. Season galette with flaky sea salt and pepper; drizzle with more oil.

Step 7

Bake galette, rotating halfway through, until crust is deep golden brown and crisp, 45–50 minutes. Let cool slightly on baking sheet. Top with finely grated lemon zest and more coarsely chopped tender herbs before serving.