Preheat oven to 375 degrees F (190 degrees C).

Whisk 2 eggs, cream, white sugar, salt, 1 pinch cayenne pepper, cinnamon, and allspice together in a bowl until batter is thoroughly combined.

Lay bread slices into batter, one at a time, and let bread absorb the mixture. Turn bread slices in batter until almost all batter has been absorbed, about 10 minutes.

Heat a large skillet over medium heat, and melt butter in the hot skillet. Cook bread slices in the hot butter until browned, 2 to 3 minutes per side. Transfer French toast slices to a baking sheet.

Lay ham slices into the hot skillet and cook until meat begins to brown, about 1 minute per side.

To assemble, place a Cheddar cheese slice on a slice of French toast, top with 2 slices of ham, and lay a Havarti cheese slice over ham.

Bake in the preheated oven until French toast pieces are no longer wet, the batter is set, and cheese has melted and begun to brown, about 20 minutes.

Place sandwiches on serving plates and top each with 2 poached eggs. Season with kosher salt and a pinch of cayenne pepper.