Heat olive oil in a large Dutch oven or pot over medium heat.
Add bacon and cook, stirring occasionally, until the bacon starts to look shiny and gets some color, for about 5 minutes.
Add onion and cook, stirring occasionally, until golden and tender, about 5 minutes.
Add thyme sprigs, garlic, black-eyed peas, and 8 cups chicken broth and bring to a simmer over medium-high.
Reduce heat to medium-low and simmer gently, skimming any foam from the surface, until beans are tender (about 35-34 minutes). Discard the thyme, then add the kale and simmer until tender.
Season with salt and pepper to taste.
To serve, drizzle beans with olive oil and top with thyme leaves and cracked pepper.