The 18-year-old was the man of the match in midweek as Chelsea ended Liverpool's FA Cup hopes with a fine 2-0 win against the runaway Premier League leaders.
He also received the accolade on Sunday as the Blues thrashed Everton 4-0 at Stamford Bridge but Lampard wants to keep the young Scot grounded.
"The minute he started training with us it was clear he could play at this level and I always had the feeling that the minute you put him in, he's just going to get better very quickly because of his quality, because of his game intelligence because of his worth ethic, because he'll put his foot in," he told the BBC.
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"Today was another example of that after Liverpool. Two really good games. I want to keep his feet on the ground. I don't think I have to worry too much at the minute because of the type of boy he is.
"He will realise that now the hard work begins and he'll be judged on a different level. He's not a newcomer any more, he needs to keep that up and of course he needs to get better because he's young.
"He has a talent for receiving the ball. He was at the base of our midfield, did everything perfectly. The minute we changed it and put him higher up, he instantly changed, took on board the information and did brilliantly as well."
Chelsea are currently fourth in the Premier League but only have an eight-point lead over 10th-placed Arsenal, having played a game more than their London rivals and Lampard believes there is plenty of time left in the race for a Champions League berth.
“Still a long way to go. Manchester United play today but are in great form. Their impactful acquisitions in January have given them a nice spring in their step. They are going to challenge. So there are quite a few teams, I don't have to name them all.
"It looks a small cushion as I sit here now but I don't feel that way. We have to get ourselves right and fight to the end."
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