It will be the biggest match of the Australian-born manager's career, and the biggest club game for his goalkeeper, Socceroo Joe Gauci, who is on loan to Port Vale from Aston Villa.
The English midlands side, who are bottom of third-tier League One, knocked out Premier League Sunderland in the fifth round to reach the quarter-finals for the first time since 1954.
They appear doomed to be relegated to the fourth tier, but Brady, who took over in January, has inspired this extraordinary Cup run.
The Newcastle-born 51-year-old left Australia to play soccer in England at 16 and stayed, forging a solid but unremarkable career in non-League and the lower divisions, and doing the same as a coach.
Chelsea, two-time European champions, will inevitably be red-hot favourites to reach the semi-finals, but Brady will hope to pull off a stunning boilover.
The other ties are harder to call. Premier League heavyweights Manchester City and Liverpool will face off, with last year's beaten finalists City set to host what promises to be a colossal clash.
Premier League leaders Arsenal, who have won the competition a record 14 times and are still in contention for a quadruple this season, will travel to promotion-chasing Championship side Southampton.
The other tie is an all-Premier League affair. Leeds will visit West Ham, who beat Brentford 5-3 on penalties after extra time on Monday to reach their first quarter-final in a decade. The match had been locked at 2-2 after extra time.
FA CUP QUARTER-FINAL DRAW
Southampton v Arsenal
Chelsea v Port Vale
Manchester City v Liverpool
West Ham v Leeds
Ties will be played around the weekend of April 4-5.
