In a large bowl, whisk together the brown sugar and eggs until smooth, about 2 mintues by hand.
Add the vanilla and vigorously beat for one minute more.
Add the peanut butter and whisk until it comes together to form a thickened dough that's the texture of frosting, about 2 minutes. Place in the refrigerator to firm up for ~15 minutes.
Preheat the oven to 350 degrees F and line 2 sheet pans with parchment paper.
Scoop roughly 2 tablespoon sized balls onto the sheet pans, leaving about 2 inches between. Use a fork to gently press down and create a criss cross pattern on each cookie.
Bake for 14 minutes, or until lightly browned around the outsides.
Let cool completely, then dip each one half way in melted chocolate, letting the excess drip off. Place the cookies right back on the parchment paper to harden, then use a fork to drizzle with any extra chocolate.
Sprinkle generously with sea salt while still wet if desired (highly recommended). Let the chocolate harden at room temperature, or speed it up by placing the pan in the refrigerator or freezer.
Store cookies in an air-tight container at room temperature for up to 5 days or refrigerate for up to a week.