For the crust: Whisk flour, 1 tablespoon of the sugar, and salt in large bowl. Add butter and cut in with pastry knife or fingers 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 and divide into 2 pieces. Form each piece into a ball, then flatten into disks and wrap separately in plastic. Refrigerate at least 30 minutes. (This can be done 2 days ahead.) Keep chilled. Let dough soften slightly before rolling out.

For the filling: 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 chilled dough disk on floured surface to 12″ round. Transfer to 9″ glass pie dish. Trim dough overhang to 1/2″. Roll out second dough disk on floured surface to 12″ round. Using large knife or pastry wheel with fluted edge, cut ten 3/4″-wide strips from dough round.

Transfer cherries 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″. 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 50 to 60 minutes longer. Transfer pie to rack and cool completely. Cut into wedges and serve with vanilla ice cream.