This paper introduces a local linear smoother for regression surfaces on the simplex. The estimator solves a least-squares regression problem weighted by a locally adaptive Dirichlet kernel, ensuring good boundary properties. Asymptotic results for the bias, variance, mean squared error, and mean integrated squared error are derived, generalizing the univariate results of Chen (Ann Inst Stat Math, 54(2):312-323, 2002). A simulation study shows that the proposed local linear estimator with Dirichlet kernel outperforms its only direct competitor in the literature, the Nadaraya-Watson estimator with Dirichlet kernel due to Bouzebda et al. (AIMS Math 9(9):26195-26282, 2024).