Preheat oven to 425°. Line a large baking sheet with parchment paper.
Press tofu by placing on a paper towel–lined plate. Lay another paper towel over tofu; cover with a plate and an object such as a can, for weight.
Meanwhile, in a medium bowl, whisk together ¼ cup sesame oil, tamari, 2 tablespoons toasted sesame oil, garlic, lime juice and ginger.
Remove tofu from press, and dry one last time with paper towel. Cut tofu into ½-inch cubes. Add to bowl with marinade; stir to coat.
Slice broccoli into florets. Add to tofu, and stir to combine. Transfer to baking sheet. Roast about 20 minutes, until tofu and broccoli begin to crisp.
While tofu and broccoli are roasting, in a small bowl whisk together remaining tablespoon sesame oil, remaining teaspoon toasted sesame oil, rice vinegar and salt.
Bring a medium pot of water to boil, and add rice noodles; cook 2–3 minutes, until soft. Drain, and rinse briefly with cold water. Return to pot, and stir in sesame oil dressing.
Serve tofu and broccoli over a bed of noodles, and garnish with sesame seeds and a lime wedge. Serve with additional tamari and red pepper flakes on the side.