This paper studies a broadband orthogonal frequency division multiplexing (OFDM) system aided by a beyond diagonal intelligent reflecting surface (BD-IRS), where inter-connections exist among different elements such that the reflection matrix can exhibit a beyond diagonal structure. Under practical circuit structures, the reflection matrix of the BD-IRS is generally dependent on the circuit parameters (e.g., capacitance matrix for all tunable capacitors) as well as the operating frequency, which leads to couplings among the BD-IRS reflection matrices over different sub-carriers and consequently new challenges in the BD-IRS design. Motivated by this, we first model the relationship between the BD-IRS reflection matrices over different sub-carriers and the tunable capacitance matrix, and then formulate the joint optimization problem of the tunable capacitance matrix and power allocation over OFDM sub-carriers to maximize the achievable rate of the OFDM system. Despite the non-convexity of the problem, we propose an effective algorithm for finding a high-quality feasible solution via leveraging alternating optimization and successive convex approximation. Numerical results show the superiority of our proposed design over benchmark designs.