In a small bowl combine warm water (3 tbs), warm milk (3 tbs) and 1 tablespoon granulated sugar. Sprinkle 2 tsp instant yeast over the liquid and let sit 5–7 minutes until foamy. In a large mixing bowl whisk 3 cups flour with 1 tsp salt.

Make dough

Add the foamy yeast mixture, 1 whole egg, and 6 tablespoons softened butter to the flour. Mix until a shaggy dough forms, then knead by hand on a lightly floured surface or with a stand mixer fitted with a dough hook for 6–8 minutes until smooth and slightly tacky. If dough is too sticky, add up to 2 tbsp flour; if too dry add up to 1 tbsp warm water.

First rise

Lightly oil a bowl, place the dough inside, cover with plastic wrap or a damp towel and let rise in a warm place until doubled, about 1 to 1.25 hours.

Shape buns

Punch down the risen dough and divide into 6 equal pieces (roughly 95–100 g each). Shape each piece into a tight round bun by folding edges under and rolling on the counter. Place buns on a parchment-lined baking sheet, spaced about 1 inch apart for a pull-apart look or 2 inches apart individually.

Proof and egg wash

Cover the buns loosely with plastic wrap or a towel and let proof 30–40 minutes until puffy. Preheat oven to 375°F (190°C). Brush each bun lightly with the beaten egg wash (1 egg + 1 tbs water). If desired, sprinkle a little coarse sugar on top for a slight sweet finish.

Bake

Bake at 375°F (190°C) for 15–18 minutes, until buns are golden brown and sound hollow when tapped. Remove from oven and let cool on a rack while preparing the filling. Keep oven on (or reduce to 350°F / 175°C later if you prefer to reheat eggs briefly).

Filling & Assembly

Prepare eggs

Crack 6 large eggs into a bowl, add 2 tablespoons cream or milk, 1 teaspoon fine salt, 1 teaspoon freshly ground black pepper, and 2 tablespoons chopped chives (if using). Whisk until combined.

Cook eggs gently

Heat 2 tablespoons unsalted butter in a nonstick skillet over medium-low heat until melted and foamy. Pour in the egg mixture and cook, stirring continuously with a silicone spatula, until eggs are soft, slightly creamy but fully set — about 3–4 minutes. Remove from heat a moment before they look fully firm; residual heat will finish them.

Assemble buns

Slice each cooled bun horizontally about two-thirds of the way through to create a pocket. If you prefer the bun warm, you can briefly return cut buns to the oven for 2–3 minutes. Add a thin smear of the optional softened butter (and Dijon, if using) to the cut side of each bun. Spoon an even portion of scrambled eggs into the bottom half of each bun (about 1 scrambled egg per bun). Sprinkle about 2 tablespoons grated cheddar onto the eggs in each bun so the warm eggs slightly melt the cheese. Place the top bun over the filling. Repeat for all buns.

Optional reheat for melting

If cheese is not fully melted, place assembled buns on the baking sheet and warm in the oven at 350°F (175°C) for 4–6 minutes until cheese is melted and buns are warmed through.