Dhameystir weedha soo socota: Ka dib markii uu ciyaaray ciyaartoy yar oo ka tirsan kooxda Ontario Hockey League (OHL) ee Windsor Spitfires, Gleason waxaa lagu soo xulay wareegga koowaad, 23aad ee guud ahaan,
2001 NHL Entry Draft oo ay sameeyeen Ottawa Senators. Isaga oo aan awoodin inuu heshiis la gaaro qandaraaska Senators, Gleason waxaa markii dambe lagu beddelay Los Angeles Kings si loogu beddelo weeraryahanka Bryan Smolinski bishii Maarso 11, 2003.