Prepare the dough: Combine the butter and milk together in a heatproof bowl. *Or use the stove and heat the butter until melted.* Pour in a medium mixing bowl. Add one egg to the butter mixture and whisk together. Set aside.
Combine 1/2 lemon zest with white sugar in a large mixing bowl then add flour, instant yeast and salt and mix together. *If you are using active dry yeast, activate it with the wet ingredients and let it sit for 10 minutes*
Pour the wet ingredients into the dry ingredients, and stir with a silicone or wooden spoon. Mix until all the ingredients are well combined. *Or simply use a stand mixer filled with paddle attachment.*
Knead the dough: Transfer the dough to a lightly floured surface, flour your hands. Knead the dough for 4-5 minutes until the dough becomes smooth and elastic but does not stick to the fingers: If your dough is sticky, add a little more flour. *If using a stand mixer, fill it with the dough hook and knead the dough for 5 minutes on medium speed.* Roll the dough into a ball and place in a lightly greased bowl, cover and let the dough rest for about 15 minutes.
Fill the Buns: Using a rolling pin, roll the dough out in a 15x8-inch (40x22-cm) rectangle. Spread the softened butter on top. Mix the brown sugar, lemon zest and cinnamon, sprinkle it all over the dough. Then , sprinkle the currant, sultanas and chocolate chips. Roll up the dough to make a 15-inch log. Cut into 10-12 even rolls and arrange in a lightly greased 9 inch round pan or 9 square baking pan.
Rise: Cover the pan with plastic wrap or a clean kitchen towel. Put in a closed oven and let it rise for about 90 minutes or until doubled in size.
Bake the buns: Once the Chelsea buns doubled in size, preheat the oven too to 370°F (190°C). Bake for 25 minutes or until the top becomes lightly browned. If you notice the tops are turning brown very quickly, loosely tent the pan with parchment paper or aluminum foil and continue baking. When done, remove the pan from the oven and place it on a wire rack until you have finished making the icing, otherwise you can the icing as the buns bake.
Make the icing: In a medium bowl, beat the cream cheese with a hand mixer until smooth, then add the butter and one tablespoon of milk and continue beating until smooth and combined. then beat in the powdered sugar and vanilla extract.
Using an icing spatula or knife, spread the icing over the warm buns and serve immediately.