Drain the soaked black-eyed peas and place them in a pot with four cups (one quart/950 milliliters) water. Bring to a boil, then reduce the heat and simmer until tender, 25 to 30 minutes. Drain the peas, reserving the cooking liquid.

In a separate pot, heat the vegetable oil over medium heat. Add the onion, garlic, and thyme and sauté until fragrant. Stir in the black-eyed peas, rice, salt, and pepper, then pour in the reserved cooking liquid. Cover and simmer until the rice is tender and the liquid is absorbed, 20 to 25 minutes.