Step 1Chop 1 cup of the herbs and set aside.
Step 2Toss the remaining cup of whole herbs with 1 teaspoon olive oil, one-half teaspoon lemon juice and salt and pepper to taste. Set aside to garnish the finished eggs.
Step 3Whisk together the eggs and one-fourth cup creme fraiche, three-fourths teaspoon salt and one-eighth teaspoon pepper.
Step 4Heat a 10-inch nonstick skillet with 1 tablespoon olive oil and butter over medium heat. After the butter melts, add the eggs. Reduce the heat to low. Use a wooden spoon to stir the eggs in a clockwise direction.
Step 5When the eggs are cooked three-quarters of the way through with curds forming, stir in the Jack cheese. Continue to cook and stir the eggs until they are still moist with some liquid. Just before they have finished cooking, stir in the chopped herbs. Continue to cook, stirring constantly until they are very creamy and almost cooked, about 15 to 18 minutes total cooking time. Remove the eggs from the heat during the last minute of cooking and stir until they are done.
Step 6Divide the eggs among the serving dishes. Top each with a dollop of creme fraiche and garnish with the herb salad and a drizzle of olive oil to finish. Serve with home fries and grilled toast.