Heat oil in a large saucepan over medium heat. Add onion and garlic; sauté until browned and fragrant, 3 to 4 minutes. Add collard greens and red pepper flakes; cook and stir until greens are slightly wilted, 2 to 3 minutes.

Add broth, vinegar, brown sugar, liquid smoke, and paprika to the pan. Cover and bring to a simmer over medium heat. Reduce the heat and simmer until greens are very tender, 35 to 45 minutes. Strain before serving.