Prepare the Tomatoes: Bring a large pot of water to a boil. Score a small 'X' on the bottom of each tomato. Blanch the tomatoes in boiling water for 1-2 minutes, until the skins start to peel back. Immediately transfer them to an ice bath to cool.
Peel the skins, then coarsely chop the tomatoes.
Place them in a bowl and toss with 1 teaspoon (5 ml / 0.2 fl oz) of salt.
Sauté the Aromatics: Finely chop the onion and mince the garlic.
Heat the olive oil in a large skillet over medium heat.
Add the onion and garlic and sauté until softened and just beginning to turn golden, about 5-7 minutes.
Add the bay leaves and cook for another minute until fragrant.
Simmer the Sauce: Add the salted tomatoes, crushed basil, tarragon, fresh parsley, and sliced olives to the skillet.
Stir everything together and bring to a gentle simmer.
Reduce the heat to medium-low and cook, stirring occasionally, for 40-45 minutes, until the sauce has thickened considerably and the flavors have melded.
Remove the bay leaves and season with black pepper and additional salt if needed.
Prepare the Omelets: While the sauce finishes simmering, prepare the eggs. For each individual omelet, crack 2 eggs into a small bowl.
Add a splash of milk and a pinch of salt and pepper.
Whisk vigorously until the mixture is light and frothy.
Cook and Fill: Melt a small amount of butter in a non-stick skillet over medium heat.
Pour in the egg mixture and cook, gently pushing the cooked edges toward the center and tilting the pan to allow the uncooked egg to flow underneath.
When the eggs are nearly set but still slightly moist on top, spoon a generous amount of the hot Provençale sauce onto one half of the omelet.
Fold and Serve: Gently fold the other half of the omelet over the sauce. Slide it onto a warm plate and serve immediately.
Repeat the process to make the remaining omelets.
Don't rush the simmering process. Allowing the sauce to cook down for the full 40-45 minutes is key to developing a deep, rich, and complex flavor.