Chelsea rule breaches that led to action were self-reported following takeover by Todd Boehly and Clearlake Capital.
The 23-year-old Rogers, one of the Premier League's top attacking talents, has signed a contract until 2033.
Gomes arrives days after club record signing Johan Manzambi, who cost more than 50 million pounds ($67 million) from Freiburg after a stellar World Cup with Switzerland