We report a supercontinuum generation (SCG) in a waveguide that spontaneously forms without an etching process during the deposition of a core material on a preformed ${\rm{Si}}{{\rm{O}}_2}$ substructure. The mechanism of dispersion control for this new, to the best of our knowledge, type of waveguide is analyzed by numerical simulation, which results in a design rule to achieve a target dispersion profile by adjusting the substructure geometry. SCG is experimentally demonstrated with a waveguide made of ${\rm{A}}{{\rm{s}}_2}{{\rm{S}}_3}$, chalcogenide glass, which has low material absorption over the mid-IR range. A dispersion-controlled waveguide with a length of 10 mm pumped with 77 pJ pulses at a telecommunication wavelength of 1560 nm resulted in a supercontinuum that extends by more than 1.5 octaves.