How can I make fresh burger buns like Hardees?
As a starter, Hardees lists the ingredients for their Soft Baked Buns on their website. 

From there, we can gain that the main ingredients are:

All Purpose White Flour
Water
Sugar
Soybean Oil
Yeast


Additionally, it contains less than 2% of:

Wheat Gluten
Salt
Maltodextrin
Food Starch-Modified
DATEM
Xanthan Gum
Whey
Dextrin
Mixed Triglycerides
Enzymes
Ascorbic Acid
Acesulfame Potassium
Maltitol
Neotame
Turmeric
Annato


And their Egg mixture for Basting:

Whole Eggs
Citric Acid diluted with water


I would probably knock out the following:

Wheat Gluten
Maltodextrin
Food Starch-Modified
DATEM
Dextrin
Mixed Triglycerides
Enzymes
Acesulfame Potassium
Maltitol
Neotame


Which leaves as the "less than 2% of" list:

Salt
Xanthan Gum
Turmeric
Annato
Ascorbic Acid
Whey