Pocket depth (PD) is a widely used biomarker for diagnosing risk of periodontal disease (PrD). However, PD typically exhibits skewness and heavy-tailedness, and its relationship with clinical risk factors is often nonlinear. Motivated by PrD studies, this paper develops a robust single-index modal regression framework for analyzing skewed and heavy-tailed data. Our method has the following novel features: (a) a flexible two-piece scale Student-$t$ error distribution that generalizes both normal and two-piece scale normal distributions; (b) a neural network with guaranteed monotonicity constraints to estimate the unknown single-index function; and (c) theoretical \revone{support}, including model identifiability and a universal approximation theorem. Our single-index model combines the flexibility of neural networks and the two-piece scaled Student-$t$ distribution, delivering robust mode-based estimation that is resistant to outliers, while retaining clinical interpretability through parametric index coefficients. We demonstrate the performance of our method through simulation studies, and an application to PrD electronic health records obtained from the HealthPartners Institute of Minnesota. The proposed methodology is implemented in the \texttt{R} package \href{https://doi.org/10.32614/CRAN.package.DNNSIM}{\texttt{DNNSIM}}.

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