After beginning the 1922–23 season with the New Haven Westminsters in the United States Amateur Hockey Association, Burch was signed to a contract by the Hamilton Tigers on January 30, 1923. He joined a Tigers team that was last in the four-team NHL, but which improved as he did. After scoring six goals in ten games to finish that season, Burch finished in a tie for third in NHL scoring in 1923–24 with 22 points. The Tigers finished 1924–25 as the regular season champions with a record, and Burch was voted the winner of the Hart Trophy as the league's most valuable player following a 20-goal regular season.