Pari gbolohun ti o tẹle: Beukeboom ṣe ere hockey junior fun Sault Ste. Marie Greyhounds (1982<unk>1985). Lẹ́yìn tí wọ́n ti yàn án ní ìpele àkọ́kọ́ (19th overall) ti
1983 NHL Entry Draft nipasẹ awọn Edmonton Oilers, o dun ni awọn ọdọ fun ọdun meji diẹ ṣaaju ki o to darapọ mọ awọn Oilers. Nigba ti o nṣere fun awọn Oilers, o gba awọn Stanley Cup mẹta, ati pe a mọ bi olutọju ti o nira.