Cox said.
The host then expressed gratitude to Gary Davies, who temporarily managed the morning slot after the BBC dismissed the previous host, 53-year-old Mills, amid a Metropolitan Police investigation into historical allegations from 2016.
"For the past few months, a legend has been doing breakfast, and we can't not say thank you to him this morning," Cox said.
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She praised Davies for stepping into the position during a transitional period for the radio network.
"He was parachuted in. He did a proper shift.
The one, the only, you love him, Mr Gary Davies," Cox said.
The broadcast featured actor Tom Hanks as the first guest in a pre-recorded interview, who offered encouragement regarding the pressure of starting a new project.
Hanks noted that confidence typically arrives by the third day of working together.
He also reacted to a listener's story about two non-verbal brothers who utilized his Toy Story character, Woody, to improve their communication skills.
"You've got a friend in me," Hanks said.
The new program retains several segments from Cox's previous afternoon show, such as "Kids In the Car," while introducing new features titled "Surprise and Shine" and "Let's hear it for the noise."
Cox joins Chris Evans and Zoe Ball as the only former Radio 1 breakfast DJs to host the corresponding morning show on Radio 2.
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Upcoming guests for the week include Olivia Rodrigo on Wednesday and Matt Damon on Friday.
