Although invented for precision measurements of single atomic transitions, frequency combs have also become a versatile tool for broadband spectroscopy in recent years. Here, we present a novel and simple approach for broadband spectroscopy, combining the accuracy of an optical fibre-laser-based frequency comb with the ease of use of a tunable external cavity diode laser. The scheme enables broadband and fast spectroscopy of more than 4 THz bandwidth at scanning speeds up to 1 THz s(-1) at sub-MHz resolution. We use this method for spectroscopy of microresonator modes and precise measurements of their dispersion, which is relevant in the context of broadband optical frequency comb generation, having recently been demonstrated in these devices. Moreover, we find excellent agreement between measured microresonator dispersion with predicted values from finite element simulations, and we show that microresonator dispersion can be tailored by adjusting their geometrical properties.