Monken had time to wander around trashing his team before games because on Freddie Kitchens gave himself playcalling duties on Sundays.
Dorsey and the organization "couldn't come to an agreement on a future restructure of the organization", ESPN first reported.
Dorsey, who in two years turned around a roster that went 0-16 in 2017 to one loaded with talent, left after refusing to accept a reduced role in the organization offered by owner Jimmy Haslam.
Though Dorsey and the Browns ownership may have butted heads, he would be an intriguing addition to the Colts front office.
In a prepared statement, the Haslams thanked Dorsey for his efforts and applauded him for creating a foundation "we need to continue to develop and build upon".
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Cleveland failed to make the playoffs in each of his two full seasons as general manager. Baker Mayfield was Dorsey's first draft selection as Browns GM in 2018, a draft that also included No. 1 cornerback Denzel Ward and star running back Nick Chubb.
Haslam said the Browns will hire a head coach and then involve the new hire in the selection of the next general manager, according to Cabot.
"Hopefully we'll learn from our past mistakes and do a much better job", Haslam said.
Other known potential candidates for the Browns' vacancy include former Green Bay Packers head coach Mike McCarthy, Buffalo Bills offensive coordinator Brian Daboll and New England Patriots offensive coordinator Josh McDaniels.