Place the sugar and yeast in a large bowl or in the bowl of a stand mixer fitted with a dough hook. Pour in the warm water, stir, and allow the yeast to activate and bubble, 2 to 4 minutes. Add the milk, thyme, 2 sticks melted butter, 2 eggs and 2 teaspoons salt. Mix until combined. Slowly add the flour a little at a time and mix until a dough forms and starts to release itself from the sides of the bowl. Transfer the dough to a floured work surface and knead by hand for about 1 minute.
Coat a large bowl with 1 tablespoon olive oil. Transfer the dough to the bowl. Place a towel over the bowl and allow to rise in a warm place until the dough has doubled in size, 1 to 2 hours.
Preheat the oven to 200 degrees F. Coat a 9-by-13-inch baking dish with the remaining 1 tablespoon olive oil.
Flour a work surface and, using a rolling pin, roll out the dough into a rectangle roughly 18 by 24 inches. Brush the dough with the remaining 2 tablespoons melted butter, sprinkle on the remaining 1/2 teaspoon salt, and generously sprinkle with 2 cups of the grated Cheddar. Roll up the dough starting at the long side of the rectangle until it is one large log. Cut the log into 2-inch-thick pieces. Arrange the pieces tightly packed, side-by-side in the baking dish.
Bake for 30 minutes.
In a small bowl, whisk the remaining egg with one tablespoon of water.
Take the rolls out of the oven. Raise the heat to 350 degrees F. Brush the rolls with the egg wash and sprinkle on the remaining 1 cup Cheddar. Bake until the rolls are golden brown and the cheese is melted and slightly brown, 15 to 20 minutes.