Witnessing an extreme, highly efficient galaxy formation mode with resolved Ly$\alpha$ and LyC emission

J1316+2614 at z=3.613 is the UV-brightest ($M_{UV}$ = -24.7) and strongest Lyman continuum (LyC, $f_{esc}^{LyC} \approx$ 90%) emitting star-forming galaxy known, showing also signatures of inflowing gas from its blue-dominated Ly$\alpha$ profile. Here, we present high-resolution imaging with the HST and VLT of the LyC, Ly$\alpha$, rest-UV, and optical emission of J1316+2614. Detailed analysis of the LyC and UV light distributions reveals compact yet resolved profiles, with LyC and UV morphologies showing identical half-light radii of $\simeq$ 220 pc. The continuum-subtracted Ly$\alpha$ emission reveals an extended filamentary structure of $\simeq$ 6.0 kpc oriented south-north with only weak/residual flux within the stellar core, suggesting a Ly$\alpha$ "hole". J1316+2614 presents remarkably high SFR and stellar mass surface densities of log($\Sigma_{SFR}$ [$M_{\odot}$/yr/kpc^2]) = 3.47$\pm$0.11 and log($\Sigma_{M}$ [$M_{\odot}$/pc^2]) = 4.20$\pm$0.06, respectively, which are among the highest observed in star-forming galaxies. Our findings indicate that J1316+2614 is a powerful, young, and compact starburst, leaking significant LyC photons due to the lack of gas and dust within the starburst. We explore the conditions for gas expulsion using a simple energetic balance and find that, given the strong binding force in J1316+2614, a high star formation efficiency ($\epsilon_{SF} \geq 0.7$) is necessary to remove the gas and explain its exposed nature. Our results thus suggest a close link between high $\epsilon_{SF}$ and high $f_{esc}^{LyC}$. This high efficiency can also naturally explain the remarkably high SFR, UV-luminosity, and efficient mass growth of J1316+2614, where at least 62% of its mass formed in the last 6 Myr. J1316+2614 may exemplify an intense, feedback-free starburst with a high $\epsilon_{SF}$, similar to those proposed for UV-bright galaxies at high redshifts. (ABRIDGED)