Heat the milk and water until they reach 110 F (43 C).
Pour the liquids into the mixer bowl, fitted with the hook, and sprinkle the yeast.
Let the yeast soften for about 2 minutes, then add the brown sugar and 1 cup (125 g) of flour with a wooden spoon.

Add the remaining flour and fine salt to obtain a sticky dough.
Add more flour if necessary (you will probably need a few tablespoons) to form a ball of dough.
Knead until the dough is soft and elastic, and no longer sticks to your fingers, about 5-8 minutes.
Shape the dough into a ball in a lightly oiled bowl, covered with plastic wrap.
Let rise in a warm place until it doubles in volume, about 2 hours.

Preheat the oven to 400 F (200 C) and line two baking sheets with parchment paper.

Place the dough on a lightly floured surface and cut it into 4 equal pieces.
Lightly flour your hands, then roll out a piece of dough to form a 12″ (30 cm) rope.
Flatten the dough and form a rectangle 4″ wide.

Place 1/4 of the prosciutto and cheddar along the length, in the lower third of the dough, leaving a 1/2″ (1 cm) border.
Roll the lower edge of the dough over the filling and press firmly to seal, then roll up as tightly as possible to form a rope.
Cut the rope into 12 pieces and transfer them to the prepared baking sheets.
Repeat with the remaining 3 ropes.
Let rest at room temperature, uncovered, for 30 minutes (the dough will slightly rise).

Bring the water to a boil in a large pot.
Reduce the heat to a simmer and add the baking soda. Add the bretzel bites to the water in batches, turning them once, until they are slightly puffed, about 20 seconds.
Remove them from the water with a slotted spoon and place them on the prepared baking sheets.

Bake for 15 minutes or until the bretzel bites are puffed and golden.

Remove from the oven and brush the bretzel bites with melted butter and sprinkle with coarse salt, if desired.

Serve warm or at room temperature.

The bretzel bites can be prepared 2 weeks in advance and frozen (once cooled).
Thaw for 30 minutes, then reheat for 10 minutes in a preheated oven at 400 F (200 C).