Butter an 8x8-inch baking pan. Line the pan with a piece of parchment large enough to hang over two sides. Butter the paper, too, and tuck it flat against the pan. Put the chopped pecan pieces in a handy spot where you'll be working.
Combine the sugar, salt, Karo syrup, half-and-half, cream, and butter in a heavy-based 3-qt. pan, stirring with a wooden spoon over low heat until the sugar is completely dissolved. This can take a while, and it's hard to see; you should feel the texture (rub a little between your fingers or run your finger along the mixture clinging to the spoon) to be sure all the sugar is dissolved.
Turn the heat to medium and cook, stirring, until the mixture foams to a boil. Add the baking soda. Lower the heat and stir like mad. The mixture will double in volume and then gradually subside and begin to take on a golden hue. After the mixture settles a bit, put in a warmed candy thermometer. Continue to stir constantly, scraping the sides, and cook over medium-low heat until the thermometer registers just 240°F. Watch very carefully, as the thermometer will hover at 239° for a while and then move up. You must remove the mixture before it passes 240°F.
Remove the pot from the heat and take out the thermometer. Continue to stir quickly. The candy will look like a loose caramel sauce. Add the vanilla (watch out, it may sputter) and stir carefully to incorporate. Add the pecans and continue stirring quickly. Don't take your eyes off the mixture at this point. Watch and feel it as it begins to thicken, lighten in color, and become harder to stir. When it has thickened enough to leave a path on the bottom of the pan while you're stirring, it's just about ready. The moment you notice that the mixture is just beginning to lose its glossy shine, turn it out into the buttered pan. Don't wait until the mixture looks completely matte or it will be too dry when you try to cut it. If you stop stirring at the right moment, the mixture will firm up almost the second it hits the pan. Too soon, it will never be anything more than caramel (although very good caramel); too long, it will harden in the pot.
As soon as the candy cools (15 to 20 minutes), cut it into squares. It will probably have tiny bubbles on top. It may well crumble when cut. If it doesn't harden immediately, just let it sit for several hours, even overnight, and it may harden. If not, you have great caramel.