Cheesy Mushroom Galette

Name: Cheesy Mushroom Galette
Ingredients: 2 cups (250 g) all-purpose flour; 1 tsp. Diamond Crystal or ½ tsp. Morton kosher salt; ¾ cup (1½ sticks) chilled unsalted butter, cut into pieces; 1 Tbsp. apple cider vinegar or white wine vinegar; 1 lb. crimini mushrooms; 3 large shallots; 1 garlic clove; ½ cup coarsely chopped tender herbs (such as parsley and/or chives), plus more for serving; 2 Tbsp. unsalted butter; 2 Tbsp. extra-virgin olive oil, plus more for drizzling; ½ tsp. Diamond Crystal or ¼ tsp. Morton kosher salt; Freshly ground pepper; 6 oz. sharp cheddar, coarsely grated (about 1½ cups); ¼ cup heavy cream; All-purpose flour (for surface); 6 oz. mixed mushrooms (such as oyster and/or king trumpet), torn; 1 large egg, beaten to blend; Flaky sea salt; Finely grated lemon zest (for serving)