Do you think I should keep my cut covered with a bandaid or let it breathe?

Superfast healing. The real question is what not to do that slows down healing. Avoid using alcohol, peroxide, hibiclens, betadine, and bleach solutions. Inflammation (red, hot, swollen, tender) is important to healing so anti-inflammatories (motrin, advil) are bad. Cover the area with a clean dressing changed every other day, and protected from trauma. Air is bad for healing! Antibiotics do not help healing.