One way to decarbonize the heat production sector is to gradually replace natural gas with hydrogen. The design of laminar premixed burners capable of operating with both methane and hydrogen is however challenging as these fuels have drastically different burning properties, which narrows the operating range. While methane flames face limitations in stabilization due to blow-off at high power, hydrogen flames tend to be susceptible to flashback at low power. This study investigates the effects of slits symmetry breaking on the blow-off of methane flames and flashback of hydrogen flames through two-dimensional direct numerical simulations of canonical asymmetrical slit configurations, revealing uneven interactions between the main openings and smaller auxiliary slits. The equations governing the reactive flow dynamics are coupled to a heat transfer solver in the solid phase to elucidate the thermal and hydrodynamic mechanisms determining the operability limits. Viscous dissipation in the small auxiliary slits together with a substantial preheating of the fresh gases by heat redistribution through the solid phase is found to govern flame stabilization in asymmetrical geometries, improving blow-off resistance of methane-air flames. Then, flashback of hydrogen-air flame is found to be driven by the competition between (i) the mass flow rate distribution between the main and auxiliary slits, (ii) preheating of the gas through the auxiliary slits and (iii) the ability of the main and auxiliary slits to quench the flame. The interplay of these phenomena gives rise to complex behaviors, wherein asymmetrical configurations could exhibit significantly enhanced resistance to flashback compared to symmetrical geometries. This conclusion, verified for different burner thicknesses and slit spacing, may be used to guide the design of fuel-flexible laminar burners.