In shallow casserole dish, whisk eggs together with milk, vanilla, sugar and nutmeg.
Spread equal amounts of cranberry sauce on one side of four slices of bread.
Top cranberry sauce with two slices of ham, 1/2 cup cheese and 1/4 cup arugula.
Top each stacked bread slice with the remaining four slices of bread. Gently press together. In large skillet, melt butter over medium/low heat.
Dip each sandwich into milk mixture. Turn and dip the other side. Place in the hot pan and cook until each sandwich is golden brown and the cheese has melted. Remove from pan and drain on paper towels. Dust with confectioners’ sugar and serve warm.