In a large skillet over medium heat, melt 3 tbsp butter and 2 tbsp olive oil. Add garlic and sauté until golden, about 1–2 minutes.
Add chicken to the skillet and season with salt and pepper. Cook until browned and cooked through, about 7–8 minutes.
Remove chicken from the skillet and set aside.
Add lemon juice and broth to the pan. Scrape the bottom of the skillet and simmer until liquid reduces by half, 2–3 minutes.
Add remaining butter and oil, then stir in cabbage and broccoli. Cook 2–3 minutes until vegetables are just tender.
Return chicken to the pan. Stir to combine and heat through. Garnish with parsley and serve.