Thermoacoustic instability within combustion systems is heavily influenced by the thermoacoustic characteristics of the burner in conjunction with its flame. A promising strategy to mitigate these instabilities involves targeting the thermoacoustic properties of the burner itself. One innovative approach for modifying or designing the burner with its corresponding flame is grounded in the heuristic notion that the acoustic response of one flame (with burner) can be counterbalanced by the appropriately tuned response of other flames. In the case of premixed gaseous multiple Bunsen-type flames anchored to the burner deck with perforations, this concept suggests the integration of various sizes and shapes of perforations in burners. However, without prior knowledge, this approach often necessitates extensive trial and error, leading to excessive costs in the Research and Development (R&D) process. Achieving a burner design and its corresponding flame that operate thermo-acoustically stable within the combustion system, while also meeting additional requirements such as emissions, operational durability, mechanical resilience, modulation rate, and others, poses a significant challenge. In this study, we initially articulate the concept of the burner transfer matrix (TM) composition to allows us to predict the TM of complex composite burners on basis of TM of its constituting parts. Then, we establish the complete framework of burner-flame TM composition based on a tabulated library of elemental burners’ pressure drop (PD), elemental burners TM, and elemental flame Transfer Functions (TF). To illustrate this design methodology, we analyze a duct-flame-duct case study. Finally, we present a stability chart that delineates thermo-acoustically safe and unsafe combinations of segments/partitions, offering valuable insights into the R&D process of burner development. By leveraging such a stability chart and considering other operational constraints, designers can systematically achieve optimized designs.