In a small bowl, stir together the brown sugar and warm milk until dissolved. In a large bowl, combine 2 ½ cups flour, the milk mixture and the yeast. Stir until a soft dough forms. Add the remaining flour as needed. Turn the dough out onto a lightly floured surface and gently knead a few times, forming a smooth ball.
Brush the inside of a large clean bowl with olive oil. Transfer the dough to the bowl and cover tightly with plastic wrap. Allow to rise in a warm, draft-free area for about 2 hours, until dough has doubled in size and bubbles appear on the surface.
Turn out the dough onto a lightly floured surface. Divide into 4 equal pieces. Lightly dust hands and rolling pin with flour. Roll one of the four sections into a 12×4-inch rectangle. With the long side facing you, gently press ¼ of the ham and cheese into the bottom third of the dough, and roll as tightly as possible, starting with the end that has the filling. Cut into 12 1-inch pieces and transfer to a baking sheet lined with parchment paper. Repeat with remaining 3 portions of dough.
Let rest, uncovered, at room temperature for 30 minutes. Meanwhile, preheat oven to 400 degrees F.
Bring 6 cups of water to a boil. Add the baking soda and reduce heat to a gentle simmer. Boil pretzels in batches, cooking about 20 seconds each, turning once. They should be slightly puffed. Use a slotted spoon to transfer them back to the baking sheets. Bake until puffed and golden-brown, about 15 minutes. Brush warm pretzel bites with melted butter and sprinkle with salt. Serve warm.