The boomerang attack is a cryptanalysis technique that combines two short differentials instead of using a single long differential. It has been applied to many primitives, and results in the best known attacks against several AES-based ciphers (Kiasu-BC, Deoxys-BC). In this paper, we introduce a general framework for boomerang attacks with truncated differentials. We show that the use of truncated differentials provides a significant improvement over the best boomerang attacks in the literature. In particular, we take into account structures on the plaintext and ciphertext sides, and include an analysis of the key recovery step. On 6-round AES, we obtain a competitive structural distinguisher with complexity 2 87 and a key recovery attack with complexity 2 61. The truncated boomerang attack is particularly effective against tweakable AES variants. We apply it to 8-round Kiasu-BC, resulting in the best known attack with complexity 2 83 (rather than 2 103). We also show an interesting use of the 6-round distinguisher on the full TNT-AES, a tweakable block-cipher using 6-round AES as a building block. Finally, we apply this framework to Deoxys-BC, using a MILP model to find optimal trails automatically. We obtain the best attacks against round-reduced versions of all variants of Deoxys-BC.