Make the caramelized onions: Heat a pan to medium heat and melt butter with olive oil.
Chop onions into thin slices, add them to butter and oil when it begins bubbling, and stir to coat onions. Add salt to taste, garlic, rosemary, and vinegar. Continue stirring mixture.
Cook onions on medium heat until they are completely soft and browned but not burnt, about 30 minutes. Once caramelized, turn off heat and set onions aside to cool slightly.
Make the bean dip: Drain and rinse the beans, add them to a food processor along with vinegar, olive oil, honey (optional), and a pinch of salt.
Add ⅔ of the cooled caramelized onions. Blend for 30 seconds or until smooth but thick in texture, some larger pieces of onion are normal. Taste and add salt if needed.
Serve at room temperature topped with remaining onions, a generous drizzle of olive, and some more fresh chopped rosemary and sliced lemon.