The paper provides a post-Keynesian explanation of the soaring inflation experienced in the post-COVID recovery, in a way that is coherent both from the microeconomic and macroeconomic point of view. To do so, four US inflation facts are presented, discussing the stylized evidence on the nature of the price surge and its implication for income distribution. At the theoretical level, the microeconomic argument is rooted into the idea that price-making firms take into account both their costs and their profits when setting prices. To defend profit margins in the aftermath of the pandemic, the initial cost push was absorbed by first passing it to prices; in a second phase, firms particularly in more highly concentrated sectors benefited of the post-pandemic permissive pricing environment to increase their price mark-ups, leading to temporary inflation and a permanently higher price level. Starting from a simple Kaleckian pricing procedure, we model the endogenous mark-up adjustment as a logistic growth process. This microeconomic explanation is compatible with the macroeconomic notion of the `inflation barrier' introduced by Joan Robinson. For a given quantity of real output, it is shown that if profit earners defend their share of income following a cost push, this will produce a one-time price increase, with inflation becoming more persistent if the target adapts endogenously - i.e. if the aggregate mark-up changes. The paper contrast the notion of a wage-price spiral with that of a profit-price sink, arguing that sellers' inflation is a real but only temporary phenomenon, leading to a higher price level but not higher inflation rates. Accordingly, we discuss the implications for the conduct of monetary policy, critically assessing its scope and limitations.