Heat olive oil in a heavy-bottomed pan or cast iron skillet over medium-low heat. Add the thinly sliced onions, salt, and pepper.

Cook slowly, stirring occasionally, for 30-40 minutes, or until the onions are deeply golden brown and very tender. Do not rush this process; slow cooking brings out their sweetness. If they start to dry out, add a tablespoon of water.

Stir in the balsamic vinegar, if using, and cook for another 2 minutes. Remove from heat and let cool completely. This step can be done ahead of time.

In a large mixing bowl (or stand mixer bowl), combine the warm milk, granulated sugar, and active dry yeast. Stir gently and let sit for 5-10 minutes until the yeast is foamy. This indicates the yeast is active.

Add 3 cups of the all-purpose flour, 1 tsp salt, and the softened butter to the yeast mixture. If using a stand mixer, attach the dough hook. Mix on low speed (or by hand with a wooden spoon) until a shaggy dough forms.

Gradually add the remaining 1/2 cup of flour, a little at a time, until the dough comes together and is no longer excessively sticky. You may not need all of it, or you might need a tiny bit more.

Knead the dough on a lightly floured surface (or with the dough hook) for 8-10 minutes, until it is smooth, elastic, and passes the windowpane test (you can stretch a small piece thin enough to see light through it without tearing).

Lightly grease a clean bowl with a little olive oil. Place the dough in the bowl, turning it once to coat. Cover the bowl with plastic wrap or a clean kitchen towel.

Let the dough rise in a warm, draft-free place for 1 to 1.5 hours, or until doubled in size.

Once the dough has doubled, gently punch it down to release the air. Turn the dough out onto a lightly floured surface.

Flatten the dough into a rectangle. Sprinkle the cooled caramelized onions and shredded cheddar cheese evenly over the surface. Gently fold the dough over itself and knead for 1-2 minutes to evenly distribute the fillings without overworking.

Shape the dough into a loaf and place it into a lightly greased 9x5 inch loaf pan. Cover the pan loosely with plastic wrap or a kitchen towel.

Let the loaf rise in a warm place for another 45-60 minutes, or until it has nearly doubled in size and looks puffy.

Preheat your oven to 375°F (190°C) during the last 15 minutes of the second rise.

Bake the bread for 35-45 minutes, or until the crust is golden brown and the internal temperature reaches 200-210°F (93-99°C).

If the top browns too quickly, you can loosely tent it with aluminum foil.

Once baked, remove the bread from the oven and let it cool in the pan for 5-10 minutes before turning it out onto a wire rack to cool completely before slicing. This helps prevent a gummy texture.