Make the crust. Combine the flour, salt and butter in a large bowl. Gently cut the butter into the flour using 2 sharp knives in a crisscross motion until the butter is about the size of small peas. (The lighter the hand, the flakier the crust.) Slowly drizzle in 1/2 cup ice water and fold into the mixture with a fork until the dough comes together-you want to work the dough as little as possible. Add more water, 1 tablespoon at a time, if the dough is too dry. Turn the dough out onto a floured surface and divide into 2 balls. Form each into a disk, wrap in plastic and refrigerate at least 30 minutes. Meanwhile, trim the rhubarb and cut into 1/2-inch pieces. Hull the strawberries and slice. Combine the rhubarb, strawberries, granulated sugar, flour, cinnamon, salt, vanilla and lemon juice in a large bowl; toss.
Prepare the bottom crust. Preheat the oven to 425 degrees F. Roll out 1 disk of dough on a floured surface into a 12-inch round. Roll the dough onto the rolling pin, then carefully unroll it over a 9-inch pie plate. Gently press the dough into the plate with your fingers. Add the strawberry-rhubarb filling. Do not trim the overhanging dough.
Make the lattice. Roll out the remaining disk of dough into a 12-inch round. With an old-fashioned crimper, sharp knife or pizza cutter, cut into 1-inch-wide strips. Lay about half of the strips on the pie in one direction, about 1/2 inch apart. Fold back every other strip halfway. Lay a strip perpendicular to the first ones. Unfold the strips back over the new strip, then fold alternate strips back halfway. Add another perpendicular strip; unfold the strips back over it. Repeat to form a lattice. Pinch the overhanging top and bottom crusts together, trimming as needed, then fold the dough under itself and crimp with your fingers.
Bake the pie. Using your fingers, pat the lattice top with milk and sprinkle with coarse sugar. Put the pie on a baking sheet and place in the preheated oven. Bake 20 minutes, then reduce the oven temperature to 375 degrees F and bake until the crust is golden brown and the filling is bubbling, 50 more minutes. Let cool completely before slicing.