Panel quantile regression models are used in finance, econometrics, insurance, and risk management. This paper introduces a two-stage method to estimate extreme conditional quantiles in the presence of individual effects and heteroscedastic extremes. By first estimating an intermediate quantile and then extrapolating to an extreme quantile, the method provides stable results. The study establishes theoretical properties and demonstrates the method's performance through simulations. Applying the method to macroeconomic and housing price data reveals evidence of housing bubbles, common economic factors, and cross-country differences.