Graphene, in single layer or multi-layer forms, holds great promise for future electronics and high-temperature applications. Resistance to oxidation, an important property for high-temperature applications, has not yet been extensively investigated. Controlled thinning of multi-layer graphene (MLG), e.g., by plasma or laser processing is another challenge, since the existing methods produce non-uniform thinning or introduce undesirable defects in the basal plane. We report here that heating to extremely high temperatures (exceeding 2000 K) and controllable layer-by-layer burning (thinning) can be achieved by low-power laser processing of suspended high-quality MLG in air in "cold-wall" reactor configuration. In contrast, localized laser heating of supported samples results in non-uniform graphene burning at much higher rates. Fully atomistic molecular dynamics simulations were also performed to reveal details of oxidation mechanisms leading to uniform layer-by-layer graphene gasification. The extraordinary resistance of MLG to oxidation paves the way to novel high-temperature applications as continuum light source or scaffolding material.