Season the game hens generously on all sides with salt.
Warm the olive oil in a Dutch oven set over medium-high heat until shimmering but not smoking. Add the game hens and brown on all sides, about 4 to 5 minutes per side. Transfer the game hens to a large plate and set aside.
Reduce the heat to medium, add the leeks to the pot, and cook, stirring occasionally, until slightly softened and translucent, about 1 to 2 minutes. Add the bacon and cook until golden brown and fragrant, about 5 to 7 minutes. Stir in the garlic and cook until fragrant, about 1 minute. Add the cabbage, lightly salt and toss to coat with the bacon fat. Cook until the cabbage is slightly wilted, 2 to 3 minutes.
Increase the heat to high, add the wine and bring to a boil. Scrape the bottom of pan with a wooden spoon to loosen any brown bits. Stir in the chicken broth and thyme. Lower the heat to medium and place the game hens and any accumulated juices on top of the cabbage. Bring to a simmer over medium heat, cover and cook for 20 minutes or until the juices run clear and the hens register 165 F on an instant-read thermometer.
Transfer the game hens to a carving board and cover with foil to keep warm. Add the cannellini beans to the pot, stir to incorporate with the cabbage and warm over low heat. Season to taste with salt and pepper. Sprinkle with parsley.
Using a carving knife, cut the game hens in half through the breastbone. Serve with the cabbage, beans and bacon.
Makes 2 to 4 servings