Chef John's Monte Cristo Benedict

Ingredients

• 2 large eggs
• ¼ cup heavy whipping cream
• 1 tablespoon white sugar
• 1 pinch salt
• 1 pinch cayenne pepper
• ¼ teaspoon ground cinnamon
• ⅛ teaspoon ground allspice
• 4 thick slices day-old French bread
• 1 tablespoon butter
• 8 thin slices cooked ham
• 4 slices Cheddar cheese
• 4 slices Havarti cheese
• 8 poached eggs
• 2 teaspoons chopped fresh chives
• or to taste
• 1 pinch kosher salt