Heat broiler. If using a broilerproof rimmed baking sheet, set rack 4 inches from heat; if using a broiler pan, set rack so that top of pan is 4 inches from heat. In a large bowl, whisk together vegetable oil, soy sauce, and 1 teaspoon salt. Add carrots; toss to coat.
With a slotted spoon, transfer carrots (reserving bowl with marinade) to sheet (or pan). Push carrots to one side.
Working with a few pieces at a time, add tofu to marinade in bowl; turn gently to coat, and transfer to sheet. Reserve bowl with marinade.
Broil until carrots are tender and tofu is browned, turning tofu halfway through cooking time and tossing carrots occasionally (more frequently toward end of cooking time), 30 to 35 minutes.
Transfer tofu and carrots to reserved bowl with marinade. Add scallions, vinegar, and sesame oil; toss gently to combine.