In this study, a novel type of air-cooled heat sink is proposed, which consists of several layers of mini-channels. In this design, the hydraulic diameter is smaller than in conventional types of heat sinks, such as plate-fin heat sinks, and consequently, the achievable heat transfer rates are higher. To predict the cooling performance of this heat sink, an innovative analytical method is proposed, results from which are complemented by an extensive number of numerical simulations of the simultaneously developing flows, thermally and hydrodynamically, inside a rectangular channel of the heat sink. The results of the analytical method are compared against two- and three-dimensional simulations and good agreement is found, while from the simulations, correlations are proposed for the Nusselt number in these flows. Finally, the performance of the proposed heat sink is compared to a plate-fin heat sink. This comparison reveals that entropy generation in the latter is around 27% higher than in the former, and suggests a promising advantage of the proposed heat sink design.