Research on chemotherapy, heart surgery, and vaccines has indicated that the risks and benefits of a treatment could vary depending on the time of day it is administered. A challenge with performing studies on timing treatment administration is that the optimal treatment time is different for each patient, as it would be based on a patient's internal clock time (ICT) rather than the 24-hour day-night cycle time. Prediction methods have been developed to determine a patient's ICT based on biomarker measurements, which can be leveraged to personalize treatment time. However, these methods face two limitations. First, these methods are designed to output predictions given biomarker measurements from a single tissue sample, when multiple tissue samples can be collected over time. Second, these methods are based on linear modelling frameworks, which would not capture the potentially complex relationships between biomarkers and a patient's ICT. To address these two limitations, this paper introduces a recurrent neural network framework, which we refer to as LassoRNet, for predicting the ICT at which a patient's biomarkers are measured as well as the underlying offset between a patient's ICT and the 24-hour day-night cycle time, or that patient's dim-light melatonin onset (DLMO) time. A novel feature of LassoRNet is a proposed variable selection scheme that minimizes the number of biomarkers needed to predict ICT. We evaluate LassoRNet on three longitudinal circadian transcriptome study data sets where DLMO time was determined for each study participant, and find that it consistently outperforms state-of-the art in both ICT and DLMO time prediction. Notably, LassoRNet obtains a median absolute error of approximately one hour in ICT prediction and 30 to 40 minutes in DLMO time prediction, where DLMO time prediction is performed using three samples collected at sequential time points.