Set a medium bowl in a large bowl of ice water. In another medium bowl, whisk the egg yolks with 1/2 cup of the sugar until pale, about 3 minutes.
In a medium saucepan, combine the cream, milk, salt and remaining 1/4 cup of sugar and bring to a simmer, whisking until the sugar is completely dissolved. Whisk the hot cream mixture into the beaten egg yolks in a thin stream.
Transfer the mixture to the saucepan and cook over moderately low heat, stirring constantly with a wooden spoon, until the custard is thick enough to lightly coat the back of the spoon, about 4 minutes; don’t let it boil. Pour the custard through a fine-mesh strainer into the medium bowl in the ice water. Let cool completely, stirring frequently. Refrigerate the custard until very cold, at least 1 hour.
Pour the custard into an ice cream maker with flavorings, if using, and freeze according to the manufacturer’s instructions. Transfer the frozen custard to a plastic container, cover and freeze until firm, at least 3 hours.
Whisk eggs, half and half, pumpkin, cinnamon, vanilla extract, pumpkin pie spice, and walnuts together in a bowl. Soak one slice of bread at a time in the pumpkin mixture, then place in the prepared skilled. Repeat with the remaining slices of bread. Stir the pumpkin mixture between dips to keep the walnuts from settling. Cook the bread until golden brown, about 3 minutes on each side.
Place the first seven ingredients in bread machine in order suggested by your manufacturer. Select dough cycle.
Turn finished dough out onto a lightly floured board and knead 5 to 10 times. Separate into 2 or 3 pieces. Roll with hands into strips. Braid or twist strips together. Place onto a parchment lined baking sheet. Set aside to rise in a warm place until doubled in size.
Preheat the oven to 350 degrees F (175 degrees C). Whisk together 1 egg white and the water. Brush onto the top of the loaf.
Bake in the preheated oven until deep golden brown, about 20 minutes.
In a medium bowl, mix the mayonnaise, garlic, and Parmesan cheese. In a saucepan over medium-low heat, mix the Cheddar cheese, half-and-half, and paprika. Stir constantly until melted and smooth. Pour into the bowl with mayonnaise mixture, stirring until well blended.
Place the French bread on the grill cut side down, and let it toast for a few minutes. Remove from the grill, and spread the cheese mixture on each side. Place the halves back together, and wrap the loaf in aluminum foil. Return to the grill for about 15 minutes, turning occasionally, until the loaf is heated through and the cheese mixture is hot.