In a heavy sauce pan, combine caramel ingredients, and stirring constantly, bring to a boil. Boil for 3 minutes, and remove from heat. Allow to cool to room temperature (about 30 minutes).
In a mixing bowl, whisk together flour, baking soda, salt, and cream of tartar.
In bowl of stand mixer, combine cooled caramel mixture with ½ cup butter. Beat until mixture lightens in color and gets fluffy. Add eggs and vanilla, and beat another 2 minutes.
Fold in dry ingredients, just until combined. Do NOT over-mix. Add pecans, if desired.
Divide dough in half, and shape into two logs about 1 ½ to 2 inches in diameter. Wrap the dough in plastic, and refrigerate several hours or overnight.
Line a baking sheet with parchment, and preheat oven to 375 degrees F.
Slice chilled dough into disks about ¼ inch thick (try to make them as uniform as possible), and place them about 2 inches apart on the prepared baking sheet. Sprinkle tops with coarse sea salt, and bake at 375 degrees for 8-10 minutes. Cool in pans for 2 minutes, and then remove to cooling rack.