Why brown it
Two things happen in that pan, not one
The toasted flavour is the reason people do it. The water
loss is the reason it changes the dough.
Butter is roughly
16–18% water. Melt it and the water stays. Cook it past melting and
that water boils off while the milk solids toast and turn nutty, so what
goes into your bowl is less butter than you started with, and drier butter
too. Butter is 16-18% water. Cold or creamed butter locks that water in fat crystals where the flour cannot reach it; melted butter frees it, so gluten forms before baking.
Drier
fat means a stiffer dough that spreads less, which is why browned-butter
recipes so often ask for an extra egg yolk, a splash of milk, or a longer
rest. Those are not flourishes. They are putting back what the pan
removed.
The browned-butter group reads chewier, darker, and thicker than the creamed and melted recipes on chew, colour, and thickness, and lands in between them on sweetness. These are different
authors' recipes sorted by method, not one dough baked three ways, so read
it as a pattern rather than a promise: some of it is the browning, and some
of it is that cooks who brown butter tend to write a different recipe in
other ways too. The full four-way comparison, including cut-in butter, is
on what melted, softened or browned butter
changes.
One recipe puts the water straight back.
The Food Lab's Chocolate Chip Cookies from
www.seriouseats.com calls for “1 standard ice cube (about 2 tablespoons; 30mL frozen water)”
whisked into the hot butter. That is not for flavour. It replaces what
browning just cooked away, and it is the clearest sign that the water
loss is real and worth compensating for.