Is there anything in the Sunnah to indicate that it is not allowed to eat again shortly after eating?

We do not know of any hadith from the Prophet (blessings and peace of Allah be upon him) which indicates that it is not allowed to eat again shortly after eating. This is more akin to the words of doctors, and in fact is something that is known from their words.

But the Messenger (blessings and peace of Allah be upon him) urged us to eat and drink little, and told us that a person should not eat more than he needs. He (blessings and peace of Allah be upon him) said: “The son of Adam does not fill any vessel worse than his stomach. It is sufficient for the son of Adam to eat a few mouthfuls, to keep him going. If he must do that (fill his stomach), then let him fill one third with food and one third with drink, and leave one third for breathing.” Narrated and classed as sahih by al-Tirmidhi (2380); also classed as sahih by al-Albani in Sahih at-Tirmidhi.

Ibn al-Qayyim (may Allah have mercy on him) said:

The Prophet (blessings and peace of Allah be upon him) told us that a few mouthfuls are sufficient to keep him going, and that does not undermine his strength or weaken him. If he does eat more than that, then let him eat what is will fill one third of his stomach, and leave the second third for water and the last third for breathing. This is what is most beneficial for the body and the heart. For if the stomach is filled with food, it leaves little room for drink, and if he follows the food with drink, it will leave little room for breathing, and will cause him to become exhausted and tired from carrying it, like one who carries a heavy load. This is in addition to what that causes of corruption to the heart (in a spiritual sense), making one find it physically difficult to do acts of worship, and provoking desires which must be fulfilled. So filling the stomach with food is harmful both spiritually and physically.

End quote from Zad al-Ma‘ad (4/17).

As-Safarini (may Allah have mercy on him) said in Ghidha’ al-Albab (2/110):

The one who eats must allocate one third [of his stomach] for food, one third for drink and one third for air, in compliance with the instructions of the Prophet (blessings and peace of Allah be upon him), who was compassionate and sincere to all people, and that guided us to what is beneficial in both religious and worldly terms… al-Harith ibn Kaladah, the physician of the Arabs, said: “Healthy eating is the best medicine and corpulence is the main cause of illness.” Al-Harith also said: “What kills people and dooms predators is eating again shortly after eating before having digested properly.”

And Allah knows best.