Roughly chop the chocolate and set aside. Reserve about 1/5 of the chocolate to be put on top of the cookies for a nicer presentation (optional).
Preheat the oven to 350°F / 180°C / Gas Mark 4, no fan (if you have a convection oven reduce the temperature by about 20° or follow the manufacturer's instructions).
Add the softened butter to the mixing bowl, beat it for a minute (you can use a hand mixer or a stand mixer, if using a stand mixer use a paddle-shaped attachment).
Add the peanut butter and beat it with butter for a minute.
Add both types of sugar. Mix over medium-low speed for about 3 minutes, until light, fluffy, and well-combined.
Add the egg and vanilla extract, mix over medium-low speed for another 2-3 minutes, until well-combined. Scrape the sides of the bowl from time to time with a spatula.
Stir the flour with baking soda and salt. Add the flour mixture to the mixing bowl. Mix over low speed until half combined (you should still see flour here and there and the dough will be rather clumpy).
Add the chopped chocolate to the bowl. Mix until the chocolate is evenly distributed and the flour fully incorporated (but don't mix for too long).
Scoop the dough with a 3-tablespoons-ice cream scoop or just with your hands/spoon (optionally, you could weigh your dough, each ball of dough should weigh 47g). Place on a baking sheet lined with parchment paper about 2 inches (5 cm) apart, don't flatten the cookies.
Bake the cookies in batches for about 14 minutes. The baking time may slightly vary depending on the oven. The cookies should have golden edges but will be completely soft. While the cookies are still soft and hot, put 2-3 chunks of chocolate on top of each cookie, and press them slightly in. Sprinkle the cookies with flaky sea salt.
Take the baking tray out of the oven and let the cookies cool on the baking sheet for about 15 minutes (they will continue to set up from the residual heat on the pan after they come out of the oven). Always use a cold baking sheet for the next batch of cookies.