Center a rack in the oven and preheat the oven to 350°F. Line a large baking sheet with parchment paper.

Melt the butter in a heavy 1 1/2-quart saucepan over medium-low heat and heat just until it begins to turn a delicate brown, about 8 minutes. Remove from the heat and pour into a medium bowl. Be careful to watch the butter and turn off the heat when it turns brown, or else you will burn it. When you heat the butter, it will boil and bubble and stay a bright yellow for a long time. At some point, it stop bubbling loudly and instead takes on a frothy texture. You can hear the sound difference in the cooking too. It is at this point that the you'll see the liquid begin to brown. As soon as you see most of the liquid turn brown, turn off the stove. See the photo in the post to see what it should look like. When it cools, the froth dissipates and you can see that the whole melted butter liquid is a deep brown like in the photo.

Set the butter aside to cool slightly, about 15 minutes. Meanwhile, sift together the flour and baking soda onto a piece of waxed paper; set aside.

Add the vanilla to the butter, and then stir in half of the sugar, blending well before adding the remaining sugar. Add the flour mixture in two additions, blending well after each addition. Set the mixture aside for 10 minutes to allow the flour to absorb the ingredients.

To shape each cookie, measure about 1 ½ teaspoons of the dough, or scoop up the dough with an ice cream scoop about 1 1/4 inches in diameter. The dough will be very crumbly. Mash it between your palms to form balls, and place on the prepared baking sheet, spacing them about 1/2 inch apart.

Bake the cookies until they puff up, crack a bit on top, and are golden on the bottom, 15 to 18 minutes. Transfer the baking sheet to a wire rack and let the cookies cool on the pan for about 5 minutes. Using a metal spatula, transfer the cookies to a wire rack to cool completely.