In a large bowl or stand mixer bowl, combine 3 cups all-purpose flour, 2 tsp instant yeast, 3 tbs granulated sugar and 1 tsp fine salt. Whisk to distribute.

Warm 1 cup water to about 105–115°F (warm to the touch). Whisk together the warm water and 3 tbs whole milk, then add to dry ingredients along with 3 tbs softened unsalted butter and 1 large egg. Mix with a dough hook or wooden spoon until a shaggy dough forms.

Knead the dough on medium speed or by hand for 6–8 minutes until smooth and slightly tacky. If dough is too sticky, add up to 2 more tbs flour. The dough should be elastic and hold together.

Place dough in a lightly oiled bowl, cover with plastic wrap or a damp towel and let rise in a warm draft-free spot until doubled in size, about 60–75 minutes.

Roll and shape buns

Turn the risen dough onto a lightly floured surface and roll into a rectangle roughly 12 x 9 inches. Spread 4 tbs softened butter evenly over the dough. Mix 4 tbs granulated sugar with 1 tsp cinnamon (if using) and sprinkle evenly over the buttered dough.

Starting at the long edge, tightly roll the dough into a log. Pinch seam to seal. Using a sharp knife or dental floss, cut the log into 8 equal pieces. Place each piece cut-side up into a lightly greased muffin tin or small bun pan, leaving space to expand.

Cover pans loosely and let buns proof until puffy and nearly doubled, 30–45 minutes. Meanwhile preheat oven to 375°F (190°C).

Egg wash and bake

Whisk 1 egg with 1 tsp water for an egg wash. Brush the tops of the buns lightly with egg wash. Bake the buns on the center rack for 18–22 minutes, until golden brown and cooked through. Remove from oven and let cool in pan 5 minutes, then transfer to a wire rack. Keep warm or cool slightly before assembling sandwiches.

Filling & Assembly

Cook the eggs

For each sandwich, heat a nonstick skillet over medium-low and add 1 tbs unsalted butter. Crack 1 large egg into a small bowl and season with about 1/4 tsp salt and 1/8 tsp black pepper. For folded or round sandwich-style eggs: whisk the egg and pour into a ring or use a small round cutter in the pan to cook an evenly shaped egg. Cook until whites are set and yolk is cooked to your liking (about 2–4 minutes for slightly set yolk). Repeat for remaining eggs (total 4). Alternatively, make thin omelets and fold to fit the bun.

While eggs finish, split each morning bun horizontally with a serrated knife. If desired, lightly toast cut sides in a pan or toaster oven until golden.

Assemble sandwiches

Spread about 1/2 tsp mayonnaise on the bottom half of each bun if using. Place one cooked egg on the bottom half, top with 1 slice of cheddar cheese (or a scant 1/4 cup shredded). Add second egg and another slice of cheese if you prefer a double-egg sandwich (suggested assembly uses two small cooked eggs per bun). Cover with the top bun and press gently. Repeat for all buns.

Finish & Glaze

If you prefer a slightly glossy, soft top, brush assembled buns lightly with remaining egg wash before serving (optional). Serve warm so cheese is melty. Store leftover buns separately from egg filling in an airtight container in the refrigerator for up to 3 days; reheat eggs gently and reassemble before eating.