In a large saucepan, heat olive oil and butter over medium heat until butter is melted.
Cook the minced garlic in the butter mixture for 2-3 minutes, until garlic starts to brown.
Add the green beans, minced onion, and pepper. Cook for an additional 3-4 minutes, until the green beans develop a bright green color (but are not cooked through).
Add chicken broth. Use enough broth so the beans are covered.
Cover saucepan and bring to a boil.
Reduce heat to medium-low. Keep the saucepan covered, but leave a vent open for steam to escape. Cook green beans for 15-20 minutes more, until most of the liquid has evaporated.