for the exact moment Luffy unlocks Advanced Observation Haki / Future Sight. Episodes 852–855 show his struggle, and 856 is the breakthrough.
Luffy thought, frustration bubbling into white-hot rage. "I know he's strong. But it's not just strength. He knows what I'm going to do before I do it."
Luffy’s eyes widened. He gasped, leaping back to create distance.
Clang.
While Episode 869 is the definitive moment for Advanced Observation Haki, several other episodes are critical to understanding its development: The first time Luffy used Advance Observation Haki. Source
They charged. This time, as equals. When does Luffy learn Advanced Observation Haki? Not in a classroom, and not through theory. He learned it in the split second he got tired of losing, unlocking the eyes to see the future to save his crew.
Unlike his basic Haki training, Luffy did not "learn" the advanced version during the two-year timeskip.