Place the butter in a small saucepan over medium heat. Let it melt, then continue cooking until it foams and the foam begins to reduce, then turns golden with nutty brown flecks at the bottom. It should smell nutty. Remove from the heat and set aside to cool slightly.
In a large bowl, whisk together the flour, baking powder, sugar and salt. In a separate bowl or jug, whisk the eggs, milk and vanilla. Slowly whisk in the warm (not hot) brown butter, including all those nutty browned flecks.
Pour the wet ingredients into the dry and gently whisk until just combined. A few small lumps are fine - overmixing makes tough pancakes. Rest the batter for 5 minutes if you have time.
Heat a non-stick frying pan over medium heat and add a knob of butter. Half fill a ladle with pancake batter, and tip into the centre of the hot butter. Cook for 2–3 minutes until bubbles appear on the surface and the edges look set, then flip and cook for another 1–2 minutes until golden.
Serve the pancakes with a dollop of yogurt, fresh berries, and perhaps a drizzle of brown butter and maple syrup.