What happens if we both connect a full path between the board edges?

If both players have valid lines connecting their respective board edges, then the player that ends the game by saying "Pavé" loses.
To win when saying "Pavé" at the end of your turn, you must have the only completed path.
You may choose not to say "Pavé" at the end of your turn, even if you have a completed path, and allow the game to continue. At this point, the best way to win is to disconnect your opponents path first, or hope that your opponent says "Pavé" to perform a win check, while you still have a completed path.