No, there is not a Hadith that says fasting in six days after Eid gives you the reward of fasting a year? Rather, the Hadith in Sahih Muslim states that fasting in Ramadan and then fasting six days in the next month (Shawal) is like fasting perpetually.
عَنْ أَبِي أَيُّوبَ الْأَنْصَارِيِّ رَضِيَ اللهُ عَنْهُ، أَنَّهُ حَدَّثَهُ، أَنَّ رَسُولَ اللهِ صَلَّى اللهُ عَلَيْهِ وَسَلَّمَ قَالَ: مَنْ صَامَ رَمَضَانَ ثُمَّ أَتْبَعَهُ سِتًّا مِنْ شَوَّالٍ، كَانَ كَصِيَامِ الدَّهْرِ
Umar bin Thabit narrated (Sayidina) Abu Ayyub al-Ansari (May Allah Most High be pleased with him) informed him that the Messenger of Allah ﷺ said, ‘A person who observed the fast of Ramadan and then followed it with six (fasts) of Shawwal, will be as if he fasted perpetually.’ (Sahih Muslim, 1164 – 204).
It is important to narrate Hadith with the correct wording. Although the Hadith above is understood to be referring to the reward of a years fasting, the actual wording is perpetual fasting. The Hadith is authentic without doubt, since all Musnad Hadith in Bukhari and Muslim are considered to be Sahih.
جَمِيع مَا حكم مُسلم بِصِحَّتِهِ من هَذَا الْكتاب فَهُوَ مَقْطُوع بِصِحَّتِهِ وَالْعلم النظري حَاصِل بِصِحَّتِهِ فِي نفس الْأَمر وَهَكَذَا مَا حكم البُخَارِيّ بِصِحَّتِهِ فِي كِتَابه وَذَلِكَ لِأَن الْأمة تلقت ذَلِك بِالْقبُولِ سوى من لَا يعْتد بِخِلَافِهِ ووفاقه فِي الْإِجْمَاع (صيانة صحيح مسلم)
Everything that (Imam) Muslim considered to be Sahih of this book is undoubtedly Sahih, and it leads to contemplative certainty due to its authenticity. Likewise, everything that (Imam) al-Bukhari considered to be Sahih in his book (is Sahih), because the Ummah has accepted that reality (the authenticity of Bukhari and Muslim) with agreement. With the exception of a few, whose acceptance or disagreement does not affect the consensus. (Imam Ibn Salah, Siyanatu Sahih Muslim).
In the Hanafi Madhab it is preferred not to fast six consecutive days. Rather, one should space the six days out. For further details, see the link below.
And Allah Most High Knows Best.
-Answered by Shaykh Noorud-deen Rashid (04.05.2022)
Do you have to fast six days in a row for Shawal?