Since its first observation in the 1970s, neutrino-induced neutral-current single positive pion production (NC1$\pi^+$) has remained an elusive and poorly understood interaction channel. This process is a significant background in neutrino oscillation experiments and studying it further is critical for the physics program of next-generation accelerator-based neutrino oscillation experiments. In this Letter we present the first double-differential cross-section measurement of NC1$\pi^+$ interactions using data from the ND280 detector of the T2K experiment collected in $\nu$-beam mode. We compare the results on a hydrocarbon target to the predictions of several neutrino interaction generators and final-state interaction models. While model predictions agree with the differential results, the data shows a weak preference for a cross-section normalization approximately 30\% higher than predicted by most models studied in this Letter.