Place the potatoes and garlic in a steamer basket fitted over a large pot of boiling water. Cover and steam until the potatoes are knife-tender, 12 to 15 minutes. Warm the chicken broth in a small pot on the stove or in a glass container in the microwave.
Remove the steamer basket and drain the water from the large pot. Transfer the potatoes and garlic to the pot, add the oil, salt, pepper, and broth and mash until smooth.