Chop 9 tablespoons of butter and measure out 1/2 cup of cream. Set them aside so they're ready when needed later.
In a clean, heavy-bottomed 3-quart pot, pour in 1/2 cup of water. Carefully add 2 1/2 cups of sugar to the center of the pot, avoiding the sides to prevent sugar crystals from forming. Add 1/4 teaspoon of cream of tartar.
Turn the heat to medium-low and stir gently with a long wooden spoon. Keep the heat steady at medium-low while the sugar dissolves completely into the water, stirring occasionally.
This process will take several minutes, so be patient. Avoid increasing the heat—you want all the sugar fully dissolved before moving on to the caramel-making stage.
Once you see bubbles covering the surface and are sure all the sugar has dissolved, remove the wooden spoon from the pot.
Increase the heat to medium-high. Do not stir, but stay close by. The mixture will continue to bubble and rise along the sides of the pot, helping to dissolve any sugar crystals that may have formed.
As it boils, the color will gradually shift from clear to yellow, eventually deepening into a dark amber shade.
The time it takes for the sugar to caramelize will vary (anywhere from 8-12 minutes) depending on factors like the burner strength and the weight of your pot. However, don't focus on timing.
The caramel is ready when two things happen: 1) it reaches a deep amber color, and 2) you see the first wisps of smoke rising from the pot.
Do not remove it from the heat until you see those smoke wisps. Even if the color seems dark and you're worried about burning, trust that if there's no smoke yet, it's not burning.
Once the caramel has reached this point, remove it from the heat. Carefully add the chopped butter using a clean wooden spoon.
The mixture will sputter, and it’s extremely hot, so be cautious. Once all the butter has melted, gradually pour in the cream. It will sputter again, so take your time.
Stir in 1 1/2 teaspoons of kosher salt. If you're using table salt, start with 1 teaspoon, and feel free to add more to taste if desired.
Allow the caramel to cool in the pot for about 30 minutes so it can thicken before pouring it over the completely cooled brownies. Let the caramel cool completely before serving.
If you're okay with messy brownies, feel free to serve them at room temperature; the caramel will gently drip over the edges when sliced (that’s how I prefer them!).
For a neater presentation, keep the brownies chilled in the fridge. Before slicing and serving, let them sit at room temperature for about 15 minutes.