Flame blade says that "You evoke a fiery blade in your free hand". For practical purposes, lets call your free hand "Left" and your cast hand "Right" (if indeed you receive the blade in the hand you didn't cast with)
After casting, would you be able to equip a shield in your right hand while still maintaining concentration of the spell? ie Cast with right hand, blade appears in left hand, equip shield into right hand.
You can use your action to make a melee spell attack with the fiery blade. On a hit, the target takes 3d6 fire damage.
Does using your action for the spell (to make the melee spell attack) require a free hand? Would it be restricted by using a shield?