My advice is to leave well alone. The blister will shrink down in size on its own.
Once you burst a blister, you will be exposing airborne germs to enter into the wound. It could then become infected.
I scalded my belly with hot water accidentally and it was quite bad, even though I had a T-shirt on. It did form a blister and was very sore, but eventually it went down itself and slowly healed. After healing the red mark will fade eventually, but lasts a very long time. It was about 2 years ago I scalded myself and I still have a small red mark where it was.
Best wishes.
A cold wet washcloth or a washcloth wrapped around an ice pack will help with the burn. Don't put anything on it like butter or vaseline. And don't poke it with a needle!
If the burn is deep or large, see a doctor.
A cold compress would only have any effect if it was done immediately and applied for a minimum of 10 minutes. It will have no benefit days after the event.