This tournament is a round robin, meaning every player/ team plays every other player/team exactly once. That requires 28 matches in total, which can be played in 7 rounds of 4 matches.
The rounds can be played as a single round per night over 7 weeks, all rounds in a single night or any other combination or order as long as each game in the bracket gets played. This also allows makeup games to be played whenever convenient.
A standard single or double elimination tournament can be played after all of the round robin matches are played where the results of the round robin establish the seeding. The seeding could be like tennis, where the goal is to have the top players meet in the later rounds, or any other seeding standard.