Make the crust: Preheat oven to 350°F. Stir together graham cracker crumbs, sugar, and salt in a bowl. Add butter and stir with a fork until well-combined. Press into bottom and up the sides of a broiler-proof 9-inch pie plate. Bake until edges are just barely golden, 6 to 8 minutes. Cool on a wire rack.

Increase oven temperature to 400°F. Pierce potatoes several times with a fork. Bake on a rimmed baking sheet until tender in the center when pierced with a knife, 1 hour to 1 hour 20 minutes. Carefully slice potatoes in half lengthwise and let stand until cool enough to handle.

Reduce oven temperature to 375°F. Scoop potato flesh from skins into a food processor; discard skins. Process until smooth, about 1 minute. Measure out 1 ½ cups puree (save remaining puree for another use).

Whisk together eggs, brown sugar, and granulated sugar in a bowl. Whisk in half-and-half, butter, and vanilla. Whisk in pureed potatoes, salt, cinnamon, ginger, nutmeg, and cloves. Pour into piecrust. Bake, until filling is set around edges (will still jiggle slightly in center), 35 to 40 minutes. Let cool completely on a wire rack, about 3 hours. Pie may be chilled at this point, if desired.

Before serving, preheat oven to broil with oven rack 5 inches from heat source. Wrap piecrust tightly with aluminum foil. Scatter marshmallows over top of pie, and broil until golden brown, rotating pie halfway through for even browning, 1 to 1 ½ minutes.