Step 1Heat the oven to 375 degrees.
Step 2In a medium saucepan, heat 1 tablespoon oil over medium heat. Sweat the onion, minced garlic, red pepper flakes and paprika for 3 minutes. Stir in the peppers, thyme and rosemary sprigs and continue to cook for 5 minutes to marry the flavors. Add the tomatoes, sugar, 1/4 teaspoon salt and several grinds of pepper, and simmer until the peppers are soft and the sauce has thickened, about 30 minutes (try to leave the tomatoes intact).
Step 3When the tomato mixture is almost ready, cook the chard: In a large cast-iron skillet heated over medium-high heat, melt the butter along with the remaining 3 tablespoons oil. Add the smashed garlic, Swiss chard, a generous 1/4 teaspoon salt and several grinds of pepper and saute until the chard is wilted, about 4 minutes. Remove and discard the rosemary and thyme sprigs from the tomato mixture, then spoon it over the cooked chard. Make 6 little indentations and break the eggs into them. Sprinkle over half of the thyme, parsley and tarragon leaves, as well as half of the oregano.
Step 4Place the skillet in the oven and bake until the yolks are just set, about 8 minutes. Sprinkle over the remaining herbs and serve immediately.