@BetoOnSecurity In professional, the highest recorded score in which both teams score is prime and total score is also prime is... 3-2.
All solutions of this problem will take the form of x-2 where x is the lower member in a pair of twin primes. Since there aren't many twin primes less than 100, it was pretty easy to brute force here: https://www.pro-football-reference.com/boxscores/game_scores_find.cgi