Heat 2 tablespoons oil over medium-high heat in a large nonstick skillet. Cook the onions until soft, about 5 minutes. Add the potatoes, 1/2 teaspoon salt and the pepper. Add 2/3 cup water and cover the skillet. Reduce the heat to medium-low and cook until the potatoes are tender, about 15 minutes. Scrape the potato mixture into a large bowl and cool, stirring occasionally, about 30 minutes. Wipe out and reserve the skillet. The potato mixture can be made 6 hours ahead. Keep at room temperature.
Combine the eggs, garlic, olives, parsley, thyme and remaining 3/4 teaspoon salt in another large bowl. Fold in the cheese and the cooled potato mixture.
Preheat the broiler. Add the remaining 2 tablespoons oil to the same skillet and heat over medium-high heat. Pour in the egg mixture, spreading evenly. Cook, uncovered, until the sides are set and the top edge begins to brown, about 5 minutes. Run a flexible heatproof rubber spatula around the sides of the pan to loosen the omelet. Cover tightly; reduce the heat to medium-low and cook until the center is almost set, about 5 minutes.
Broil the omelet until the top is fully cooked and begins to brown, 3 to 4 minutes. Let the omelet rest for 2 minutes. Loosen the omelet again around the sides and underneath as far as possible. Slide the omelet onto a platter and garnish with parsley and thyme sprigs. Cut into wedges. Serve warm or at room temperature with Quick and Easy Romesco Sauce.
Quick and Easy Romesco Sauce:
Combine the bell peppers, oil, tomato paste, almonds, vinegar, garlic, paprika, salt and pepper in a blender or food processor. Blend until the sauce is almost smooth, scraping down the sides of the bowl occasionally. If the sauce is too thick, blend in the reserved roasted bell pepper juice, 1 teaspoonful at a time, to thin out the sauce. Transfer the sauce to a small bowl. The sauce can be made 1 day ahead. Cover and chill. Bring to room temperature before serving.