In a wide, shallow, heavy-bottomed pot (or rondeau), heat 3/4 cup of the olive oil over medium-high heat until hot. Season the guinea fowl with salt and pepper. In batches if necessary, sear the fowl skin-side down until golden brown, 3 to 4 minutes, then flip and cook for an additional 3 to 4 minutes. Remove to a plate.
Add the remaining 3 tablespoons olive oil, then add the white wine and garlic cloves and cook, stirring constantly, 4 to 5 minutes. Add the thyme and cook for an additional minute. Using a wooden spoon, scrape the bottom of the pot to remove the renderings. Bring to a simmer and reduce the heat to medium low. Cook for 3 minutes to slightly reduce the sauce.
Return the guinea fowl to the pot and cook for 20 minutes, rotating the fowl every 5 minutes, until cooked through. Transfer the guinea fowl to a serving platter. Ladle the sauce over the meat and serve.