How to Make It
Preheat the oven to 350° and position a rack in the center of the oven. In a heavy-bottomed skillet, melt the butter over medium-low heat. Continue to cook; the butter will foam and then begin to turn golden, then nut brown; whisk occasionally. When the butter is nut brown, immediately add the brown sugar, whisk, and then carefully add the water to loosen. Bring the mixture to a boil and continue simmering until a candy thermometer reads 225°F. (If you don’t have a candy thermometer, cook until the mixture smells caramelized and starts to darken.) Slowly add the heavy cream (the mixture will bubble rapidly) and whisk until smooth. Allow to cool for at least 10 minutes. Stir in the vanilla extract.
Place the prebaked pie shell on a rimmed baking sheet. In a separate bowl, lightly whisk the eggs and yolks together with the salt. Set aside. In the bowl of a food processor fitted with the blade attachment, blend the pumpkin puree with the allspice, cinnamon, ginger, cloves, molasses, and lemon juice until smooth. With the machine running on low, stream the brown-butter butterscotch through the food processor’s feed tube and process until combined. Stream in the egg mixture, followed by the milk and carrot juice; blend until smooth, stopping once or twice to scrape down the sides with a rubber scraper.
Strain the filling through a fine-mesh sieve into a separate bowl, pressing through with a rubber scraper. Pour into the prebaked shell. Bake on the middle rack of the oven for 45 to 55 minutes, rotating 180 degrees when the edges start to set, 30 to 35 minutes through baking. The pie is finished when the edges are set and puffed slightly and the center is no longer liquid but still quite wobbly. Be careful not to overbake or the custard can separate; the filling will continue to cook and set after the pie is removed from the oven. Let to cool completely on a wire rack, 2 to 3 hours. Serve slightly warm, at room temperature, or cool.