The visitors, meanwhile, have failed to win a trophy since lifting the FA Cup in 1995, but they may be focused on securing their survival in the Premier League.
Match preview
Having posted their lowest finish in the English footballing pyramid for 20 years - when they were promoted from the Conference in 2003 - by finishing 18th in League Two last season, Doncaster appear to be in serious danger of falling outside of the Football League after failing to win any of their opening five league games.
Just one point from five matches sees them lie rock-bottom of the table after losing 2-1 away to Milton Keynes Dons on Saturday, with Grant McCann already facing serious pressure after joining as manager for his second spell at the club in May.
Rovers' only real cause for celebration all season came in the opening round of this competition, with George Miller's brace helping them come from behind to defeat Championship side Hull City - who McCann guided to the League One title in 2021 - and set up a box-office clash against Everton in the second round.
McCann will be urging his players to put on a real show in front of Sky Sports' television cameras at the Eco-Power Stadium on Wednesday, with a positive performance giving them a good chance of turning things around in the league across the coming weeks and months.
Likewise, Everton's campaign has began in disastrous fashion, with the Toffees sitting bottom of the Premier League after losing each of their opening three matches without scoring a single goal.
Having finished 16th and 17th in the previous two seasons, one of England's most decorated football clubs are clearly trending in the wrong direction, bringing their 69-year stay in the top flight ever closer to its conclusion.
Sean Dyche's side were on the brink of gaining their first point of the season against Wolverhampton Wanderers at Goodison Park on Saturday, but substitute Sasa Kalajdzic scored with only three minutes of normal time remaining to claim a precious victory for his side in a clash that already represented a potential relegation six-pointer.
Ahead of visiting another team without a point in the shape of Sheffield United on Saturday, Dyche will welcome the respite this competition can provide them in midweek, with his players likely to receive a much required confidence boost should they claim a victory as expected.
Doncaster Rovers EFL Cup form:
W
Doncaster Rovers form (all competitions):
L
W
L
D
L
L
Everton EFL Cup form:
Everton form (all competitions):
D
W
W
L
L
L
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Team News
McCann may look to reward the same starting XI that defeated Hull in the previous round, meaning Miller is set to lead the hosts' line alongside Luke Molyneux.
Deji Sotona may start in a more advanced role than he is generally accustomed to, potentially allowing him to display his excellent dribbling ability on a more regular basis.
Everton, meanwhile, will likely restore Michael Keane to their defence after dropping the England international against Wolves following his calamitous performance against Aston Villa.
Youssef Chermiti could make his first start for the club following his debut from the bench on Saturday, while fellow new signing Beto could be involved at some point after completing his £30m move from Udinese.
Alex Iwobi, Dominic Calvert-Lewin, Andre Gomes, Dwight McNeil, Dele Alli, Jack Harrison and Seamus Coleman are all expected to remain sidelined through injury.
Doncaster Rovers possible starting lineup:
Lawlor; Bailey, Olowu, Anderson, Maxwell; Sotona, Broadbent, Biggins, Roberts; Molyneux, Miller
Everton possible starting lineup:
Virginia; Godfrey, Keane, Branthwaite, Mykolenko; Garner, Onana; Dobbin, Maupay, Danjuma; Chermiti
We say: Doncaster Rovers 1-2 Everton
With both teams yet to win in the league this season, confidence could be at an extremely low ebb for both sets of players, but the visitors should have enough about them to avoid being on the wrong end of an upset.
Injuries have ravaged Dyche's squad and will limit his ability to rotate as much as he may have wanted, but they have plenty of attacking players desperate to prove themselves and should be able to grab a couple of goals against their lowly-ranked opponents.