Heat 1 tablespoon olive oil in a large skillet over medium heat until the oil shimmers.
Add 1 cup kale (stems removed and roughly chopped), 2 cups collard greens (stems removed and roughly chopped), 2 cups mustard greens (stems removed and roughly chopped), and 1 cup Swiss chard (stems removed and roughly chopped) to the skillet. If the skillet is crowded, add the greens in batches, pressing them down so they contact the pan.
Cook the greens, stirring or tossing occasionally, until they begin to wilt and reduce in volume, about 3–5 minutes.
Stir in 3 cloves garlic (minced), 1 cup chickpeas (cooked and rinsed), 1/2 teaspoon kosher salt, and 1/4 teaspoon ground black pepper.
Continue to cook, stirring occasionally, until the garlic is fragrant and the greens are fully wilted and heated through, about 2–4 minutes.
If excess liquid collects in the skillet, drain or spoon off the liquid to your preference.
Serve the garlic-braised greens and chickpea salad hot.