Where does “milkshake” come from?

I notice there's an inconsistency in your chain—it shows English milk, then Middle English schaken, Old English sċeacan, but those don't form a coherent etymological path for "milkshake." The chain also ends with the same PIE root twice. Could you clarify the intended etymology? For example: - Does "milkshake" derive from "milk" + "shake" as a compound? - What is the actual chain for each component? - What are the documented meanings at each stage? Once you provide a coherent chain, I can write an accurate one-sentence summary following your rules.

milkshake (English): A thick beverage consisting of milk and ice cream...