Combine the lemon balm, olive oil and garlic in the bowl of a food processor. Process until combined but still slightly chunky. Reserve.
Fill a large pot with lightly salted water and bring to a rolling boil over high heat. Once the water is boiling stir in the spaghetti and return to a boil. Cook the pasta uncovered, stirring occasionally until the pasta has cooked through but is still firm to the bite, about 12 minutes. Drain well.
Combine the tomatoes, onions and dried mushrooms in a large frypan over medium-high heat. Bring to a simmer, breaking up the tomatoes with the back of a spoon. Cook until the tomato liquid has partially evaporated, about 15 minutes. Stir in the arrowroot, salt and the prepared lemon balm pesto. Simmer to blend the flavours and thicken the sauce, about 5 minutes.
Serve the lemon balm pesto sauce over the hot cooked spaghetti.