use DBIish; 

role DB-Connection {
    has $.dbh;

    submethod TWEAK( :$dbh! ) { 
            $!dbh = $dbh.isa(DBIish) ?? $dbh !! DBIish.connect( |%$dbh
+ ); 
    }
}
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Notes: The ! after $dbh, in the signature of the TWEAK method makes $dbh argument required. The colon indicates it is a named argument, rather than a positional one.
Attributes are always read only by default.
The |%$dbh does two things. The % "dereferences" (rather re-containerizes) the Hash in $dbh. The | then flattens that hash into named subroutine arguments.