Apple Pie with Rosemary and Honey

In a food processor, mix flour, 2 sticks butter, 1/4 cup brown sugar, and salt until butter flakes are about 1/2 inch. Add milk, 1 tablespoon at a time, until dough just begins to form a ball. Divide dough into 2 balls, flatten slightly, and wrap tightly in plastic wrap. Refrigerate for at least 1 hour or up to 1 day. In medium bowl, combine apples with remaining brown sugar, spices, and tapioca. Add honey and stir until combined. Preheat oven to 400 degrees F. On a lightly floured surface, roll out 1 ball of chilled dough to an 11-inch round. Fit into a 9-inch deep-dish pie plate. Trim edges of dough, leaving a 1-inch overhang all around. Add apple filling to pie pan and dot with remaining 2 tablespoons butter pats. To make lattice piecrust: On a lightly floured surface, roll out second ball of dough to a 1/8-inch-thick 13- by 5-inch rectangle. Using a pizza wheel, fluted pastry cutter, or a paring knife, cut 10 half-inch-wide strips. Weave strips to form a lattice pattern. Brush egg wash over lattice crust. Sprinkle lattice with coarse sugar. Bake pie on middle rack of oven. Place a cookie sheet on bottom rack to catch any juices that may bubble over. After 30 minutes, remove pie from oven and cover edge of crust with foil to prevent burning. Continue to bake until juices are bubbling, 50 minutes more. Transfer pie to a wire rack to cool for 45 minutes. Serve warm with a slice of Cheddar, if desired.