Brown has started the season on top form for Celtic and Scotland after noticeably improving his game last season despite fitness problems.
And former Celtic midfielder Grant, who coached the team under Tony Mowbray, feels Brown is benefiting from a new-found maturity.
Grant, who was launching the fourth year of Tesco Bank's coaching partnership with the Scottish Football Association at Hampden, said: "Even last season and this season he has been outstanding. When Tony made him captain, I thought at the time it was the wrong call because I didn't think he was disciplined enough to even be in the side at the time.
"So what he has done in the last 18 months has been outstanding and he deserves all the plaudits he gets.
"I think he is getting better and better. His game has matured, I think he has matured as a person and that is showing in his all-round performances.
"I was down at the Wembley game and I thought he was outstanding against top-quality international players there.
"I have seen him in European games playing exceptionally well and I think that's all down to having a more disciplined approach to his all-round game. He is reaping the rewards and he has been fantastic."
Brown has only been booked once in 12 games this season and Grant admits he is "100 per cent" surprised to see the turnaround in temperament.
The former Scotland international could not have foreseen the transformation when he left Celtic in March 2010 just after Brown had been sent off in an Old Firm derby for wrestling with Kyle Lafferty in a game in which Maurice Edu scored the only goal in the dying seconds.
Grant said: "We said to him at half-time he had to be careful and he got sent off in the first kick of the second half near enough.
"That really let us down badly because we lost a goal in the 94th minute.
"That lack of discipline, he had it in his locker. He is still going to have his moments but I think his all-round game has improved because his discipline has improved.
"His anxiety to run forward and just run all the time, he has settled down a bit. He is playing the position more and looks a more mature, all-round player.
"Joe Ledley has to take credit for that because the position the plays alongside him seems to have settled him down a bit.
"He is a massive, massive player for Celtic in and out of possession, his passing is a lot better and his selection of pass is a lot better, his pass completion rate looks a lot more than it used to be.
"All these things make him a more rounded player.
"He deserves to wear that armband and if he keeps playing the way he is, he could end up being a Celtic great."
Grant believes Brown and in-form striker Anthony Stokes will be key players as Celtic embark on their Champions League group campaign against Milan in the San Siro on Wednesday.
The pair both missed large chunks of last season through injury and Grant feels they can help offset the loss of Gary Hooper, Victor Wanyama and Kelvin Wilson.
"They have lost a goalscorer who is maybe going to get you 30 goals," Grant said. "That could be a blow and I do think they are a little bit weaker than they were last year all over the pitch because of the players they have lost.
"But they can hang on to the experience of last season, you have Scott Brown playing as well as he has probably played for Celtic, Stokes is playing exceptionally well and they are going to be big, big players."
SFA youth ambassador Grant backed the Tesco Bank Challenge, which has now engaged 61,112 children in Scotland.
"We got the bug at four of five because the only thing we had was a ball," the 48-year-old said. "If we can get the ball out to kids then hopefully they will catch the bug and we can produce Scotland stars of the future."