Finely chop your vegetables. Add olive oil or butter to a small skillet and saute your veggies until softened.
Meanwhile, split your English muffins, start any needed prep for your meat, and slice the cheese.
Start toasting your English muffins. Add your eggs and half & half to a small bowl and measuring cup and whisk thoroughly.
Add the beaten eggs to the skillet along with the veggies. Allow the mixture to set undisturbed until somewhat firm on the bottom. Flip half the mixture over so it resembles an omelet. If it doesn't flip perfectly, no worries.
Flip the egg mixture over one more time. Meanwhile, pull the English muffins, cheese, and meat together. Top the bottom of each muffin with cheese, meat, and half the egg mixture.