Wash hands with soap and water.
If using fresh onions, scrub them with a clean vegetable brush under running water. Place onions in a saucepan; add water to cover. Bring to a boil; reduce heat and simmer 6 to 8 minutes or until onions are just tender. If using frozen onions, read the package instructions for directions on how to cook the frozen onions safely.
Melt 2 tablespoons of butter in a heavy saucepan over low heat; add flour, stirring until smooth. Cook 1 minute, stirring constantly. Gradually add half-and-half; cook over medium heat, stirring constantly, until sauce is thickened.
Add salt, pepper, ground mustard, and nutmeg. Add 1 cup of the shredded cheddar cheese; stir until cheese melts. Gently stir in drained onions.
Pour onion mixture into a lightly greased 1-quart baking dish. Sprinkle evenly with remaining ½ cup shredded cheddar cheese.