Make the caramel: (The caramel can be made a day ahead and softened again in the microwave for easy pouring. It really doesn't take a lot of time though.)
Pour the sugar into a medium saucepan and heat to medium-high. Using a rubber spatula, stir the sugar just occasionally, scraping the bottom of the pan. At first the sugar will be lumpy, but will soon melt and start turning brown. Don't stir too often, just enough to keep everything evenly cooking. After about 5 minutes, the sugar should be fully melted and become a rich amber color.
At this point, remove the pan from the heat and immediately add the butter. the mixture will start to bubble quite rapidly. That's just what you want.
Using your spatula or a whisk, stir the butter and sugar until they melt together almost completely. This can take 1-2 minutes.
Next, pour in the heavy whipping cream and sea salt. If you're making regular caramel, then add 1/2 teaspoon of salt and no more. If you're making a salted caramel version, add anywhere from 1 teaspoon to 1 1/2 teaspoons depending on your liking.
Stir constantly now until the sauce is all mixed together and is smooth. Place back on the stove at medium-high heat again for about 20 seconds more. Remember to keep stirring.
Remove from heat and serve immediately for a hot option over ice cream or let cool at least 15 minutes for room temperature use.
Over several minutes, the caramel will become harder to stir. For a manageable sauce, just heat in microwave for 10-15 seconds and stir again.
Make the brownies:
While the caramel is cooling, preheat oven to 350F degrees. Very lightly, butter an 8"x8" baking dish. Now cut two 8"x 16" pieces of parchment paper. Place one piece in the dish with edges coming up the sides. Very lightly butter the bottom of that piece of paper (this is so the second piece will adhere to it). Place the second sheet of parchment paper in dish coming up the other sides of the dish. Again, very lightly butter the parchment so all surfaces the brownies will touch are covered (I can't stress enough, use a light hand). Set aside.
In a medium bowl or stand mixer bowl, stir together the melted butter and sugar until mixed well. Add eggs, vanilla extract, salt and optional coffee, mixing on medium speed of the stand mixer or quickly by hand. Once combined fold in cocoa powder and flour. If using, toss in the bittersweet chocolate chips. Cook's note: The chocolate chips make this doubly chocolatey and are delicious, but your brownies will be SUPER gooey.
Pour about halve the brownie batter into the baking dish, spreading evenly to the edges and up the sides of the pan a little.
Pour 1/2 cup (more or less, depending on your liking) of the salted caramel over the brownie batter, spreading evenly but not quite to the edges.
Spoon the rest of the brownie batter over the top, sealing as much of the caramel in as you can.
Bake the brownies at 350F degrees for 30 minutes. Then decrease temperature to 325F degrees and continue baking for an additional 10 minutes or until a knife insterted in the center comes out clean. Reducing the heat for the last few minutes helps the brownies to bake evenly from edges to center without the edges getting over-cooked.
Remove from oven and allow to cool completely (about 1 hour). Cut into 9 squares.
Serve with extra caramel drizzled over each brownie, if desired. Also, sprinkle with coarse salt (just a little).