Biomarker development is currently a high priority in neurodevelopmental disorder research. For many types of biomarkers (particularly biomarkers of diagnosis), reliability over short periods is critically important. In the field of autism spectrum disorder (ASD), resting electroencephalography (EEG) power spectral densities (PSD) are well-studied for their potential as biomarkers. Classically, such data have been decomposed into pre-specified frequency bands (e.g., delta, theta, alpha, beta, and gamma). Recent technical advances, such as the Fitting Oscillations and One-Over-F (FOOOF) algorithm, allow for targeted characterization of the features that naturally emerge within an EEG PSD, permitting a more detailed characterization of the frequency band-agnostic shape of each individual's EEG PSD. Here, using two resting EEGs collected a median of 6 days apart from 22 children with ASD and 25 typically developing (TD) controls during the Feasibility Visit of the Autism Biomarkers Consortium for Clinical Trials, we estimate test-retest reliability based on the characterization of the PSD shape in two ways: (1) Using the FOOOF algorithm we estimate six parameters (offset, slope, number of peaks, and amplitude, center frequency and bandwidth of the largest alpha peak) that characterize the shape of the EEG PSD; and (2) using nonparametric functional data analyses, we decompose the shape of the EEG PSD into a reduced set of basis functions that characterize individual power spectrum shapes. We show that individuals exhibit idiosyncratic PSD signatures that are stable over recording sessions using both characterizations. Our data show that EEG activity from a brief 2-min recording provides an efficient window into characterizing brain activity at the single-subject level with desirable psychometric characteristics that persist across different analytical decomposition methods. This is a necessary step towards analytical validation of biomarkers based on the EEG PSD and provides insights into parameters of the PSD that offer short-term reliability (and thus promise as potential biomarkers of trait or diagnosis) vs. those that are more variable over the short term (and thus may index state or other rapidly dynamic measures of brain function). Future research should address the longer-term stability of the PSD, for purposes such as monitoring development or response to treatment.