Step 1Heat the oven to 350 degrees. In a large bowl, whisk together the eggs with the rum and vanilla. Whisk in the sugar and brown sugar until incorporated, then whisk in the flour. Fold in the walnuts and half of the chocolate chips until evenly incorporated.
Step 2In a medium saucepan, melt the butter. Remove from heat and stir in the remaining chocolate chips until melted and combined.
Step 3Fold the melted butter and chocolate mixture into the large bowl with the rest of the filling. Pour the filling into the par-baked pie shell. Place the pie in the oven and bake just until the pie sets (it will jiggle just slightly when tapped), 30 to 45 minutes.
Step 4Remove and cool completely before serving.