They can regenerate from NOTHING, thanks to the Elixir.
The elixir of immortality created by Eirin Yagokoro by using Kaguya Houraisan's power over eternity. Whoever drinks this forbidden elixir will cease to age, never become sick, and never stay dead no matter how many times they are killed. Their body is still able to be damaged no different than before but will regenerate from any injury. Even if their body is destroyed it will reform, as the elixir causes the soul itself to become immortal. Unfortunately pain is still experienced like normal as well, causing certain immortals to feel the pain of death many hundreds of times only to resurrect each time whether they want to or not.
Because this elixir is not only forbidden, but only known to be able to be made by one single medical genius (Eirin) and even then only with the help of Kaguya, it is understandably rare. Only 3 people are believed to have ever drank it- Kaguya, Eirin, and Fujiwara no Mokou. However some people believe it's unclear whether Eirin ever drank the elixir herself. In the past this elixir has been used as a bribe, but everyone bribed with it were mysteriously murdered before they could drink it.
Reportedly the ingredients of the elixir remain in the liver of whoever drinks it. Eating the raw liver of an immortal is said to grant the same immortality to whoever eats it.