In a medium saucepan, whisk together brown sugar, flour, and salt. Set pan over medium heat and slowly whisk in milk. Cook mixture until it forms a thick, smooth sauce, about 8 minutes.
Spoon ¼ cup warm sauce into egg yolks and whisk to combine. Slowly whisk egg mixture into saucepan and continue whisking for about 2−3 minutes until filling is thick and smooth. Remove from heat and stir in butter and vanilla. Pour filling into prebaked piecrust and set aside.
Adjust oven rack to middle position and preheat oven to 350 degrees.
For meringue, beat egg whites and cream of tartar with an electric mixer on high speed until foamy. One tablespoon at a time, add sugar and beat until stiff peaks form. Add vanilla and beat until blended. Spread meringue evenly over filling, sealing to edge of crust.
Bake pie until meringue is nicely browned, about 10 minutes. Let cool completely, then place in refrigerator. Chill at least 2 hours before serving.
Pecan Shortbread Crust
In a food processor fitted with blade attachment, pulse pecans until finely chopped. Remove from processor and set aside.
Add flour and sugar to processor and pulse lightly to combine. Slice chilled butter into ½-inch slices and add to processor. Pulse until mixture is crumbly.
Add pecans to mixture and, with motor running, pour water and vanilla through food chute. Process just until dough forms a ball. Transfer dough to a floured surface, shape into a ball, and chill 30 minutes.
Roll dough between two sheets of plastic wrap to ¼-inch thickness. Transfer to a 9-inch pie plate, removing plastic wrap. Trim excess dough from edges; cover and chill for 30 minutes.
Preheat oven to 350 degrees. Prick bottom and sides of dough with fork and line with parchment paper. Fill pie with dried beans or pie weights and bake for 20 minutes or until edges began to brown lightly. Remove beans and paper; return crust to oven. Bake until crust is dry and evenly browned on bottom, about 8−10 minutes. Set aside to cool while preparing filling.