Little red dots (LRDs) may be powered by supermassive black holes (SMBHs) accreting above the Eddington limit. Spectroscopy of LRDs often shows absorption troughs blueshifted by ∼100-300 rm km s⁻¹, implying a slow wind. This is puzzling: super-Eddington disks drive much faster winds, ≳10³-10⁴ rm km s⁻¹. We argue that LRDs are super-Eddington SMBHs viewed off-axis and engulfed in a slow wind that covers most sight-lines; the fast wind escapes near the poles. Trapped light puffs the inner disk into a quasi-spherical envelope that launches the slow wind. The wind may be ``photon-tired'', meaning the light only barely unbinds it. Such marginally unbound winds lead to fountain flows, where some gas escapes and the rest falls back. We model the envelope with idealized, spherically symmetric ``marginally unbound'' (v∼ v_(rm esc)) and ``photon-tired'' winds. We feed these profiles into the radiative transfer code Sirocco to study how the wind reprocesses the light from an accreting 10⁶ M_⊙ SMBH. We describe most of the wind as a ``Balmer cocoon'' – a Compton-thick region in which depletion of Balmer continuum photons (hν >3.4 rm eV) rather than Lyman continuum photons (hν > 13.6 rm eV) keeps the gas ionized. The spectra span the range of LRD-like sources: ``little blue dots'' at lower outflow rates (∼2.5 M_⊙ rm yr⁻¹); V-shaped LRDs with a Balmer break (∼5-10 M_⊙ rm yr⁻¹); and red LRDs with full breaks (∼15 M_⊙ rm yr⁻¹). Our Balmer line profiles show P Cygni features atop broad, exponential wings, as is observed. The break is possible at lower wind densities than in LTE models because Lyman~α trapping sustains our n=2 hydrogen population and electron scattering enhances the optical depth. Our arguments are also applicable to winds from supermassive stars or quasi-stars.