Separate the kale leaves from the stalks. Discard the stalks, keeping only fresh-looking leaves that aren’t limp or yellowing. Wash them in cold water and chop into large pieces.
Place the ¼ cup of the oil in a saucepan, add the kale and sauté for 5 minutes. Add the vegetable broth and sauté for another 5 minutes.
Brush both sides of the bread with the remaining oil and toast both sides under the broiler. Rub one side of each slice with the garlic clove. Place a quarter of the kale on each slice of toast, sprinkle with salt, and season with a twist of pepper.