Put the butter in the bowl of a mixer fitted with the paddle attachment and beat on medium speed until the butter is soft and creamy. (Alternatively, you can do this and all subsequent steps by hand, working with a sturdy rubber spatula.)
Add both sugars, the salt, and vanilla extract and beat for another minute or two.
Reduce the mixer speed to low and add the sifted dry ingredients. Mix only until the dry ingredients are incorporated -- the dough will look crumbly, and that's just right.
For the best texture, you want to work the dough as little as possible once the flour is added. Toss in the chocolate pieces and mix only to incorporate.
Turn the dough out onto a smooth work surface and squeeze it so that it sticks together in large clumps. Gather the dough into a ball, divide it in half, and working with one half at a time, shape the dough into logs that are 1-1/2 inches (4 cm) in diameter. (Cookie-dough logs have a way of ending up with hollow centers, so as you're shaping each log, flatten it once or twice and roll it up from one long side to the other, just to make certain you haven't got an air channel.)
Wrap the logs in plastic wrap and chill them for at least 1 hour. (Wrapped airtight, the logs can be refrigerated for up to 3 days or frozen for I month.).
Center a rack in the oven and preheat the oven to 325 degrees F (165 C). Line two baking sheets with parchment paper and keep them close at hand.
Working with a sharp thin-bladed knife, slice the logs into rounds that are 1/2 inch (1.5 cm) thick. (Don't be upset if the rounds break; just squeeze the broken-off bit back onto the cookie.) Place the cookies on the parchment-lined sheets, leaving about 1 inch (2.5 cm) spread space between them.
Bake only one sheet of cookies at a time, and bake each sheet for 12 minutes. The cookies will not look done, nor will they be firm, but that's just the way they should be. Transfer the baking sheet to a cooling rack and let the cookies stand until they are only just warm or until they reach room temperature.