Photonic integrated Raman lasers have extended the wavelength range of chip-scale laser sources and have enabled applications including molecular spectroscopy, environmental analysis, and biological detection. Yet, the performance is strongly determined by the pumping condition and Raman shift value of nonlinear medias, leaving challenges to have a widely and continuously tunable Raman laser (e.g., over 100 nm). Here, photonic engineered Raman lasers based on chip-integrated chalcogenide microresonators are demonstrated. The home-developed chalcogenide photonic platform is of high nonlinearity, wide transparency, and low loss. The strong and broadband material Raman response has promised rich dynamics of Raman lasing. Indeed, both single-mode Raman lasing and a broadband Raman-Kerr comb, which are found engineered by tuning the dispersion of the chalcogenide microresonator, are demonstrated. The single-mode Raman laser, together with its cascaded modes, supports a gap-free tuning range over 140 nm, while the threshold power is as low as 3.25 mW. The results may contribute to the understanding of Raman and Kerr nonlinear interactions in dissipative and nonlinear microresonators, and on application aspect, may pave a way to integrated and efficient laser sources that is desired in spectroscopic applications in the infrared.