Heat 1/4 cup of the olive oil in a medium pan over medium heat. Add the garlic and cook for 1 minute, stirring. Add the parsley, thyme, lemon juice and zest, and red pepper flakes, season with salt and pepper, and cook for 5 minutes longer, stirring. Remove from the heat and set aside.
Add the eggs and milk to a medium bowl. Season with salt and pepper and whisk until combined. Set aside.
In a large pan, heat the butter and olive oil over medium-high heat. Add the eggs to the pan and cook for 4-5 minutes or until just set. Spoon the herb mixture evenly one side of the omelet and fold the other side over.
Transfer the omelet to a serving plate. Sprinkle additional parsley and freshly ground black pepper, if desired, over the top and serve immediately.