Prepare the produce and ham: thinly slice the Onions, the Red Bell Peppers and the Green Bell Pepper; mince the Garlic; roughly chop the Ripe Tomatoes; finely chop the Flat-leaf Parsley; and tear or dice the Bayonne Ham.
Warm a large heavy skillet over medium heat and add a glug of Extra-virgin Olive Oil. When shimmering, add the sliced Onions and cook, stirring occasionally, until translucent and beginning to soften, about 5–7 minutes.
Add the sliced Red Bell Peppers, Green Bell Pepper and the minced Garlic. Continue to cook, stirring occasionally, until the peppers have softened and the edges get a little color, about 8–10 minutes.
Stir in the chopped Ripe Tomatoes, then season with Piment d'Espelette, a generous pinch of Sea Salt and a small pinch of Sugar to balance the acidity. Reduce the heat to medium-low and simmer, uncovered, until the mixture is thick and jammy, about 15–20 minutes. Taste and adjust seasoning.
Make 4 small wells in the pepper-tomato mixture and crack an Egg into each well. Scatter the torn or diced Bayonne Ham over the top so it warms through as the eggs cook.
Cover the skillet and cook gently until the egg whites are set but the yolks remain soft, about 6–8 minutes (longer if you prefer firmer yolks).
Finish with a sprinkle of the chopped Flat-leaf Parsley and, if you like, a little extra Piment d'Espelette or a final drizzle of Extra-virgin Olive Oil.
Serve straight from the pan with plenty of crusty bread for dipping into the saucy peppers and runny yolks.
