Under the right conditions brown dwarfs that gain enough mass late in their lives to cross the hydrogen burning limit will not turn into low-mass stars, but rather remain essentially brown dwarf-like. While these objects, called either beige dwarfs or over-massive brown dwarfs, may exist in principle, it remains unclear exactly how they would form astrophysically. We show that accretion from AGB winds, aided by the wind Roche lobe overflow mechanism, is likely to produce a substantial population of observable overmassive brown dwarfs, though other mechanisms are still plausible. Specifically we predict that sun-like stars born with a massive brown dwarf companion on an orbit with a semi-major axis of order 10 AU will likely produce overmassive brown dwarfs, which may be found today as companions to the donor star's remnant white dwarf. The identification and characterization of such an object would produce unique constraints on binary evolution because there is a solid upper limit on the brown dwarf's initial mass.