In a medium bowl, combine milk and eggs. Spread mustard over 4 of the slices of bread. Top with ham and close sandwich with remaining bread. Pinching sandwiches together with your fingers, dip each into egg mixture to coat.
In a large skillet set over medium-high heat, melt 1 1/2 tsp (7 mL) butter. Fry 2 sandwiches at a time until golden on both sides, about 8 minutes. Place on a baking sheet. Repeat for remaining sandwiches.
Top each sandwich with cheese. Broil until bubbly and golden.
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