Brush a 9x9-inch (23x23-cm) baking dish with the olive oil.
In a medium bowl, whisk together the eggs, milk, and chives. Season with salt and pepper.
Add the egg mixture to the baking dish and bake until just set in the middle, 15-20 minutes. Cut into 9 squares.
For kids, layer one slice of cheese on top of one of the English muffin halves. Top with a frittata square. Spread the remaining English muffin half with mustard, and sandwich together.
For adults, layer one slice of cheese on top of one of the English muffin halves. Top with a frittata square, roasted red pepper slices, and arugula. Spread the remaining English muffin half with mustard, and sandwich together.