Preheat the oven to 350 degrees F (175 degrees C).

In a large bowl, whisk together the eggs, rum, and vanilla until well combined.

Whisk in the sugar and brown sugar until fully incorporated.

Whisk in the flour until the batter is smooth.

Fold in the chopped walnuts and half of the chocolate chips until evenly distributed.

In a medium saucepan, melt the butter over low heat.

Remove the saucepan from the heat.

Stir in the remaining chocolate chips until completely melted and the mixture is smooth.

Fold the melted butter and chocolate mixture into the large bowl with the rest of the pie filling.

Pour the filling into the par-baked pie shell.

Place the pie in the preheated oven and bake for 30 to 45 minutes, or until the filling is set but still jiggles slightly when tapped.

Remove the pie from the oven and let it cool completely before serving.