Preheat oven to 375°F (190°C). Line baking sheets with parchment.
Whisk flour, cream of tartar, baking soda, and salt in a bowl. Set aside.
Beat butter and 1 1/2 cups sugar until fluffy, about 2 minutes. Add eggs one at a time, then vanilla.
Mix in soy sauce and salted caramel until combined. Gradually add dry ingredients and mix until dough forms.
Scoop tablespoon-sized dough, roll in remaining 3 tbsp sugar, and place 2 inches apart on sheets.
Bake 9–11 minutes until edges set and centers look slightly soft. Cool 5 minutes on sheet, then transfer to rack.
Drizzle with extra caramel and sprinkle flaky sea salt once cooled.