Whisk flour, sugar, and salt in large bowl to blend. Add butter and rub in with fingertips until small pea-size clumps form. Add 5 tablespoons ice water; mix lightly with fork until dough holds together when small pieces are pressed between fingertips, adding more water by teaspoonfuls if dough is dry. Gather dough together; divide into 2 pieces. Form each piece into ball, then flatten into disk and wrap in plastic. Refrigerate at least 30 minutes. Do ahead Can be made 2 days ahead. Keep chilled. Let dough soften slightly before rolling out.
Position rack in lower third of oven and preheat to 425°F. Whisk 1 cup sugar, cornstarch, and salt in medium bowl to blend. Stir in cherries, lemon juice, and vanilla; set aside.
Roll out 1 dough disk on floured surface to 12-inch round. Transfer to 9-inch glass pie dish. Trim dough overhang to 1/2 inch. Roll out second dough disk on floured surface to 12-inch round. Using large knife or pastry wheel with fluted edge, cut ten 3/4-inch-wide strips from dough round. Transfer filling to dough-lined dish, mounding slightly in center. Dot with butter. Arrange dough strips atop filling, forming lattice; trim dough strip overhang to 1/2 inch. Fold bottom crust up over ends of strips and crimp edges to seal. Brush lattice crust (not edges) with milk. Sprinkle lattice with remaining 1 tablespoon sugar.
Place pie on rimmed baking sheet and bake 15 minutes. Reduce oven temperature to 375°F. Bake pie until filling is bubbling and crust is golden brown, covering edges with foil collar if browning too quickly, about 1 hour longer. Transfer pie to rack and cool completely. Cut into wedges and serve with vanilla ice cream.