Melt the butter in a medium saucepan over medium heat. Add the flour and whisk immediately and continuously for 2 minutes until the roux is smooth, bubbling and faintly golden. This cooking of the roux removes the raw flour taste. Do not rush this step.
Add the cream and milk gradually, whisking continuously after each addition to prevent lumps. The mixture will be very thick initially and will loosen as more liquid is added. Continue whisking until completely smooth.
Add the eggplant purée to the béchamel and stir thoroughly to combine. The mixture will turn a warm grey-beige colour with flecks of smokiness throughout. Cook over medium-low heat for 5 minutes, stirring frequently, until the purée and béchamel have fully integrated and the mixture is thick, smooth and hot.
Remove from heat. Add the grated cheese and stir until completely melted into the purée. Season generously with salt and white pepper. Taste at this point. The purée should be rich, smoky, creamy and well seasoned. If it seems too thick, add a splash of warm milk to adjust.