Grease and flour a 9-inch tart pan with removable sides. Preheat the oven to 350 degrees F.
Melt the butter in a bowl set over simmering water. Add 2 cups of the chocolate chips, remove from the heat, and stir until the chocolate melts. Set aside to cool completely.
In the bowl of an electric mixer fitted with the paddle attachment, beat the eggs, sugar, coffee, and vanilla on medium-high speed until light and fluffy, about 3 minutes. Stir in the cooled chocolate. In a medium bowl, combine the flour, baking powder, salt, 1 cup of the chocolate chips, and the walnuts. Fold the flour mixture into the batter until just combined. Pour into the pan and bake for 35 to 40 minutes, until the center is puffed (the top may crack). The inside will still be very soft. Cool to room temperature before removing the sides of the tart pan.
Melt the remaining 1/4 cup of chocolate chips with the heavy cream and drizzle on the tart.