Place coconut oil, olive oil, beans, onion powder, lemon juice, cider vinegar, and salt in a food processor. Blend until thick and smooth, about 1 minute. Transfer to a large bowl and chill, covered, until cold and set to a spreadable consistency, at least 1 hour.

Melt oil in a large skillet over medium-high heat. Cook onions with a pinch of salt and a few dashes pepper, stirring frequently, 3 minutes. Reduce heat to low and cook, stirring occasionally, until onions are very soft, sweet, and amber brown, about 35 minutes more. Transfer to a bowl and cool completely, about 1 hour.

Fold caramelized onions into tangy creamy cheese with a rubber spatula. Season with salt and pepper, and sprinkle with green onions.