Add 1 1/2 cups of lukewarm water and the yeast to a large bowl and allow the yeast to sit until foamy, about 5 minutes. Whisk in 1 teaspoon of salt and 1 tablespoon of the melted butter. Add the flour and stir with a wooden spoon to combine. Once the dough is shaggy, but mostly holding together, turn it out onto a lightly floured surface. Dust your hands with flour and knead the dough for about 3 minutes, adding more flour as needed to prevent sticking. Form the dough into a ball and place back inside the mixing bowl. Cover with plastic wrap or a clean kitchen towel and leave in a warm, dry place to rise until the dough has doubled in size, about 1 hour. Punch the dough down once to release some of the air bubbles. Turn the dough out onto a very lightly floured surface and divide the dough into 8 equal pieces with a pizza cutter or a sharp knife. Form each piece into a ball, and place seam side down. Let rise, uncovered, for 10 minutes.
Meanwhile, line a rimmed baking sheet with parchment paper. Preheat the oven to 400 degrees F. Bring 12 cups of water and the baking soda to a light simmer in a large pot.
Submerge 2 to 3 dough balls at a time into the simmering water using a slotted spoon. Let them sit in the water for 10 seconds per side. Carefully remove each roll, making sure to drain off as much liquid as possible and place on the prepared baking sheet. Using a sharp knife, make an X on the top of each roll. Brush the remaining melted butter on the rolls and sprinkle with a generous pinch of sea salt. Bake until golden brown, about 25 minutes. Remove from the oven and let rest for 10 to 15 minutes before slicing.
To assemble the sandwiches: Slice each roll in half. Spread softened butter on each of the cut sides of the roll, using about 1 tablespoon of butter per roll. Delicately fold and layer ham on the bottom half of each roll. Top with brie and place about a 1/4 cup of arugula on top of the ham. Sandwich with the top buns.
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