If using plum tomatoes, cut a small shallow X in the bottom of each tomato. Prepare an ice bath in a large bowl and bring a large pot of water to the boil. Add tomatoes to the boiling water and blanch until the skin starts to peel at the edges of the cuts, 10 to 20 seconds. Drain and transfer to the ice bath. Once the tomatoes are cooled slightly, use your fingers to peel off the skin and discard. Roughly chop the tomatoes; set aside.

Season the chicken pieces with salt. In a dutch oven or large pot over a medium heat, add 3 tablespoons of oil. Add the chicken to the pan, skin-side down (working in batches, if needed). Cook until the skin is golden brown, 8 to 10 minutes, then transfer the chicken to plate and set aside. Add the ham to the pot and cook, stirring occasionally, until browned and crisp, about 6 minutes. Transfer the ham to the plate with the chicken.

Add the remaining 1 tablespoon of oil to the pot and add the garlic and onions. Cook, stirring occasionally, until the onions are soft and starting to brown, about 8 minutes. Add the bell peppers and season with salt. Continue to cook, stirring occasionally, until the peppers start to soften, 8 to 10 minutes. Add the wine, reserve chopped tomatoes, the thyme, bay leave and piment d’Espelette and lower the heat to simmer. Return the chicken and ham to the pot and cover, leaving the lid slightly ajar. Cook for 20 minutes.

Remove the lid and continue cooking until the liquid is almost completely reduced, 30 to 40 minutes more. Season with salt and serve warm.