Submerge the dried peas in cool water for at least 6 hours, then drain and rinse them before cooking. The soak hydrates the peas, helping them soften more evenly in the broth.

Melt the butter in a large pot over medium-high heat, then stir in the onion, garlic, green bell pepper, and celery and cook for 3 to 4 minutes. The vegetables should begin to soften and smell fragrant as their flavors blend with the butter.

Fold in the soaked peas, pour in the chicken broth, and add the ham hock with a modest amount of salt, pepper, and cayenne. The low-sodium broth gives you room to adjust seasoning once the smoked meat has flavored the liquid.

Bring the liquid to a boil, lower the heat to a gentle simmer, and cover the pot for 30 minutes. Look for small, steady bubbles rather than a rolling boil, which can break the pea skins.

Check the peas and broth after the covered simmer, then cook uncovered for about 15 minutes if the liquid needs reducing or add more broth if the mixture is too thick. Tender peas should yield easily when pressed, while the broth should become cloudier and lightly silky rather than dry or paste-like.

Once the peas are tender, stir in the white vinegar and taste the pot, adjusting salt, pepper, or cayenne as needed. The finished flavor should feel savory and smoky with a brighter edge, not sharply vinegary.

Spoon the peas and their cooking liquid over white or brown rice, or gently combine the pea mixture with cooked rice before serving. Leave enough pot liquor to moisten the grains rather than draining it away.