In a medium bowl stir together the flour, sugar, basil, baking powder, and salt. Add egg, milk, shredded cheese, and oil. Stir just until moistened. Add batter to a preheated, lightly greased waffle iron and bake according to manufacturer's directions. When done, use a fork to lift waffle off waffle iron. Repeat with remaining batter. You should have 8 4-inch waffles.
For each sandwich, spread waffles with mustard. Layer ham, sliced cheese, and arugula on half of the waffles. Top with remaining waffles, spread side down.
Heat a griddle or 12-inch skillet over medium heat. Add butter until melted. Add sandwiches. Cook for 4 minutes or until cheese is melted, turning once.
Nutrition Facts (per sandwich): 619 cal., 35 g total fat (13 g sat. fat), 155 mg chol., 2118 mg sodium, 35 g carb., 1 g fiber, 10 g sugar, 35 g pro.
Daily Values: 17% vit. A, 1% vit. C, 41% calcium, 11% iron