In a medium to large saucepan, heat several glugs of olive oil or knobs of butter over medium heat. Add the thyme sprigs and fry until fragrant and the leaves have stopped making the popping sound, about 1 minute. Add the onion and half the garlic, then reduce the heat to low and sweat until tender, 5 to 7 minutes. Add the beans and enough water to cover by 2 to 3 inches. Bring to a boil, reduce to a simmer, and cook until the beans are completely tender and creamy, and the water is mostly absorbed. This should take 45 to 75 minutes, depending on the variety and age.

When they’re done, remove the beans from the heat and add the rest of the garlic, the salt, and buttermilk. Stir to combine, cover, transfer to the fridge, then allow the beans to cool in the buttermilk—at least 30 minutes, but ideally overnight.

Before serving, remove the sprigs of thyme (don’t worry if some leaves break up into the broth) and gently warm the beans over low heat. When the beans are warm, taste and adjust the seasoning as desired.

In a large frying pan, heat a glug of olive oil over high heat until almost smoking, then add the greens and a big pinch of salt. Sautée the greens until they are bright in color and starting to soften. Add the white wine and chile flakes (if using) and allow the wine to evaporate by half.

Reduce the heat to medium and make four divots in the greens. Drizzle more olive oil into the greens and crack an egg into each divot. Season the eggs with salt and black pepper. Cover the pan with a lid to steam the eggs—5 to 6 minutes, checking frequently toward the end so they don’t overcook.

Dish the beans into serving bowls. Scoop a nest of greens and an egg from the pan and nestle on top of the beans. Drizzle with a bit more olive oil and serve on its own, or with a thick slice of toast.