Preheat the oven to 350°F (180°C).
In a large bowl, whisk together the eggs, cream, salt, and sugar. Place the English muffin halves, cut side up, in a single layer in a baking dish or roasting pan. Pour the egg mixture evenly over the top and let the muffins soak until saturated, flipping the muffins once, about 10 minutes.
Using a fork or tongs, lift a muffin half from the batter, letting the excess liquid drip back into the bowl, and place it on a plate, cut side down. Place a cheese slice on top of the muffin, followed by 1 or 2 slices of ham. Crown with a schmear of mustard. Sandwich with another English muffin half, cut side up. Repeat with the remaining ingredients.
In a large ovenproof skillet over medium to medium-low heat, heat 1 tablespoon of butter until it melts, tilting the skillet to evenly slick the surface with butter. Carefully arrange the assembled sandwiches in the pan and cook until the bottoms are lightly browned, about 3 minutes. (It may be necessary to cook the sandwiches in batches to not crowd the skillet. Add more butter to the skillet in between batches if necessary.)
Using a wide spatula, flip the sandwiches. Slide the skillet into the oven and bake just until the cheese is melted and bubbly, about 10 minutes, flipping the sandwiches once again halfway through.
Serve right away. Don't forget to pass jam, maple syrup, and/or confectioners’ sugar at the table.