Preheat the oven to 375ºF (190ºC).
In saucepan, bring 1-1/2 cups water to boil. Stir in bulgur; cover and simmer until tender and water is absorbed, about 15 minutes.
Add arugula and cream cheese; stir until arugula is wilted and cheese is melted. Add dill, mint, lemon juice, garlic, 1/2 tsp of the salt and pepper. Stir in 1/2 cup of the cheddar.
Meanwhile, cut thin slice from stem ends of tomatoes; using melon baller or small spoon, scoop out centres. Sprinkle cavities with remaining salt. Place, cut side down, on towel to drain, about 5 minutes.
Divide bulgur mixture evenly among tomato cavities, stuffing and packing firmly. Sprinkle with remaining cheddar cheese.
Bake on baking sheet in oven until tomato skin is wilted and cheese is golden brown, about 20 minutes.