Apple Pie Doughnuts

Name: Apple Pie Doughnuts
Prep time: 30 min
Cook time: 15 min
Total time: 45 min
Cuisine: American
Ingredients: 2 packages (1/4 ounce each) active dry yeast; 1 cup warm apple cider (110° to 115° F); 6 tablespoons unsalted butter, melted until browned; 1/3 cup granulated sugar; 1 large egg plus 1 egg yolk, at room temperature; 2 Tablespoons whole milk; 3/4 teaspoon salt; 1/2 teaspoon ground nutmeg; 1 teaspoon ground cinnamon; 3 and 3/4 cups all-purpose flour, divided; 3 cups frying oil, (I used refined coconut oil, but you may use canola or vegetable oil); 5 large apples, peeled, cored, and diced; 3 and 1/2 tablespoons unsalted butter; 1/4 teaspoon salt; 1/4 cup white sugar; 1/3 cup light brown sugar, packed; 2 teaspoons ground cinnamon; 2 and 1/2 tablespoons apple cider; 1/2 tablespoon all-purpose flour; 1 cup granulated sugar; 1 Tablespoon cinnamon