Braided Coffee Cake With Apple-Ginger Filling

Ingredients

• 1/4 cup whole milk
• 2 tablespoons water
• 1 1/8 teaspoons active dry yeast
• 2 large eggs, at room temperature
• 3 tablespoons granulated sugar
• 1/4 teaspoon fine salt
• 2 cups all-purpose flour
• 3 tablespoons unsalted butter, cut into small pieces and at room temperature, plus more for coating the bowl
• 1 pound granny smith apples (about 2 medium)
• 3 tablespoons unsalted butter
• 1/2 cup packed light brown sugar
• 1/2 teaspoon ground cinnamon
• 1/4 teaspoon fine salt
• 2 tablespoons finely chopped candied or crystallized ginger
• all-purpose flour, for dusting the work surface and rolling pin
• 2 tablespoons plus 2 teaspoons whole milk, plus more as needed
• 1 1/2 cups powdered sugar, sifted
• 1/2 teaspoon vanilla extract