Make the toffee: Preheat the oven to 425° F. Line a 9x13-inch baking dish with parchment paper. See notes for doubling the toffee on a sheet pan to make extra. Lay the crackers in an even layer in the bottom of the pan and set aside.
Melt the butter and brown sugar in a medium saucepan over medium-high. Stir occasionally until the mixture comes to a boil then turn the heat down to medium and continue to boil for 4-5 minutes, without stirring.
Remove it from the heat and pour the caramel mixture over the cracker layer. Spread with a wooden spoon or rubber spatula to cover them evenly. Bake in the preheated oven for 4-5 minutes, until just bubbly.
Remove from the oven and sprinkle on the chocolate chips. Allow it to sit for a few minutes to melt and soften then use a rubber spatula to spread the chocolate over the caramel in an even layer. Sprinkle with sea salt, if desired. Transfer the toffee to the freezer while you make the cookie dough.
Brown the butter: Place butter in a medium saucepan over medium heat. Swirl the butter around to melt it and continue to stir constantly for 5 minutes. The butter will begin to bubble and foam and the butter will turn golden brown. Some of the bubbles will subside around the 5-minute mark and you'll start to see the milk solids start to toast and turn a golden brown color. It should smell buttery and nutty. Remove it from the heat immediately and allow it to cool slightly (10 minutes).
Make the dough: While it's cooling, whisk together the flour and baking soda in a large bowl. When the browned butter is slightly cooled, add the brown sugar and granulated sugar to the saucepan and whisk to combine. Whisk in the eggs and vanilla until no yellow streaks remain. Pour the wet ingredients into the flour mixture and stir with a rubber spatula until combined. It will be dry at first then start to come together.
Place the dough in the freezer for a few minutes while you break up the toffee. Remove the toffee from the freezer and use a sharp knife to chop it into small pieces. Using your hands to break it will melt some of the chocolate. Remove the dough from the freezer and add almost all of the toffee, saving some for the top of the cookies after baking. Stir well to evenly distribute the toffee into the dough then cover the dough with plastic wrap and chill for at least 2 hours, or up to 3 days. Allow the dough to sit out and soften slightly if it chills longer than 24 hours.
Preheat the oven to 350°F. Line a large baking sheet with parchment paper or a silicone baking mat.
Scoop the dough into 2 Tablespoon-sized balls and place them on a cookie sheet a few inches apart. Bake for 8-10 minutes, until the edges are golden brown and the top is just set.
Top them with reserved toffee bits and allow them to cool on the baking sheet for 5 minutes before transferring to a wire rack to cool completely.