Position racks in the top third and centre of the oven and preheat the oven to 350°F (177°C). Line two large rimmed baking sheets with parchment paper.
To make the filling: Mix the Monterey Jack cheese, feta, ricotta salata, dill, mint, parsley, eggs, and pepper together in a large bowl.
Unroll the filo and stack on the work surface. Cover the filo with a sheet of plastic wrap and a damp kitchen towel to keep it from drying out.
Keeping the remaining filo covered, place a filo sheet, with the short side facing you, on the work surface. Using a soft pastry brush, gently brush the filo with melted butter. Using a pizza wheel or a sharp knife, cut the filo into thirds lengthwise.
Place a teaspoon of the filling on the bottom of one strip, about 1” from the bottom and slightly off centre. Starting at the right bottom corner of the strip, fold the filo over to cover the filling to meet the opposite long side of the strip. Lightly press down on the filled filo to spread the filling.
Fold the filo up, then over to the other side, continuing in a “flag-folding” fashion to form a filled triangle-shaped borag. Transfer to a prepared baking sheet. Repeat with the remaining two filo strips. Continue with the remaining filo, filling, and butter, placing the borags about an inch apart on the baking sheets.
Brush the tops of the triangles with more melted butter. Bake until the borags are golden brown and crisp, about 15 minutes.