Combine warm water (about 105–115°F / 40–46°C) with granulated sugar in a small bowl. Sprinkle active dry yeast over the water, stir lightly, and let sit 5–10 minutes until foamy.

Mix dough

In a large bowl or stand mixer fitted with a dough hook, combine 4 cups of the all-purpose flour and salt. Add foamy yeast mixture, softened butter, and olive oil. Mix on low to combine, then gradually add remaining flour (about 2 cups) until a shaggy dough forms.

Knead

Knead the dough by machine for 6–8 minutes (or by hand 8–10 minutes) until smooth, elastic, and slightly tacky but not sticky. Add a teaspoon of flour or a little water if necessary to reach the right consistency.

First rise

Lightly oil a large bowl. Place dough in the bowl, turn to coat with oil, cover with a damp cloth or plastic wrap, and let rise in a warm draft-free spot for 1 to 1¼ hours, until doubled in size.

Filling

Prepare garlic butter

While dough rises, melt 4 Tbsp unsalted butter in a small saucepan over low heat. Add minced garlic and cook gently 1–2 minutes until fragrant (do not brown). Remove from heat and stir in chopped parsley, Dijon mustard (if using), and black pepper. Set aside to cool slightly.

Prepare ham and cheese

Thinly slice or julienne the deli ham if not already sliced. Combine grated cheddar and shredded mozzarella in a bowl and toss to distribute evenly.

Assemble filling layers

After the dough has doubled, punch down and turn it out onto a lightly floured surface. Divide dough into two equal portions for two loaves or keep whole for one large loaf. Roll each portion into a rectangle roughly 12 x 8 inches and about 1/4 inch thick. Brush the surface evenly with about half of the garlic butter (reserve some for brushing later). Spread a thin layer of ham evenly over the dough, then sprinkle about half of the cheese mixture over the ham. Starting from the long side, tightly roll the dough into a log. Pinch seam to seal. Repeat for second portion if making two loaves.

Optional cut-and-braid (alternative assembly)

For a pull-apart presentation, after rolling logs, make 8–10 equal-length cuts across the top of each log (not all the way through), or slice into 8–10 segments and place cut-side up in a parchment-lined loaf pan, slightly angled so cheese and ham peek out between layers.

Second rise

Place assembled logs into a greased 9x5-inch loaf pan (or two smaller pans) seam-side down. Cover loosely with plastic wrap and let rise 30–45 minutes until puffy and nearly doubled.

Topping / Finish

Preheat oven and egg wash

Preheat oven to 375°F (190°C). Beat 1 egg with 1 tablespoon water to make an egg wash.

Brush and top

Brush the risen loaf(es) with the egg wash, then brush with a little of the reserved melted butter and garlic butter mixture for extra flavor. Sprinkle the top with flaky sea salt and the remaining chopped parsley. If you used the cut-and-braid method, brush the exposed cheese with butter to promote browning.

Bake

Bake in the preheated oven for 25–35 minutes for a single large loaf (or 20–30 minutes for smaller loaves), until the crust is deep golden brown and the internal temperature reaches about 190–200°F (88–93°C). If the top is browning too quickly, tent with foil for the last 5–10 minutes.

Rest and finish

Remove from oven and immediately brush with remaining melted butter. Let the loaf rest in the pan 10 minutes, then transfer to a wire rack to cool another 10–15 minutes. Slice and serve warm so cheese is gooey.