Prepare the Cucumbers: Peel, seed, and finely dice the cucumbers.
In a large skillet, melt 3 tablespoons / 42g (1.48 oz) of the butter over medium heat.
Add the diced cucumber and sauté, stirring frequently, for about 20 minutes, or until all the liquid has evaporated and the cucumbers are tender.
Create the Béchamel Base: In a medium, heavy-bottomed saucepan, melt the remaining 2 tablespoons / 28g (0.99 oz) of butter over low heat.
Whisk in the flour to form a paste (a roux) and cook for 2-3 minutes, stirring constantly to avoid browning. Gradually whisk in the heated milk until smooth. Increase the heat to medium and continue stirring until the sauce thickens enough to coat the back of a spoon.
Season with salt and white pepper to taste.
Blend the Sauce: Remove the thickened sauce from the heat.
In a blender or food processor, combine the sauce, the cooked cucumber, the flesh of the avocado, lemon rind, and lemon juice.
Blend until completely smooth and creamy.
Final Touches and Serving: Pour the puréed sauce back into the saucepan.
Stir in the minced onion and gently reheat over low heat until just warmed through. It is crucial not to let it simmer or boil, as this can alter the fresh avocado flavor.
Serve immediately for the best taste and texture.
Add a handful of fresh herbs like dill, chives, or parsley during the blending step for an extra layer of freshness.