Slice tofu into thirds lengthwise so you have 3 even slabs. Stack the slabs on top of each other and slice through them lengthwise to make 3 even columns, then slice across to make 5 even rows.
Lay tofu evenly on a plate lined with a couple of sheets of paper towels. Layer more paper towel on top of the tofu and place another plate on top of the stack. Press out as much water as possible and let sit for at least 30 minutes.
Preheat the oven to 400 degrees F (200 degrees C). Line a baking sheet with parchment paper.
Transfer pressed tofu to a mixing bowl and drizzle with sesame oil. Lightly toss to combine. Sprinkle with cornstarch and ground chipotle chile pepper. Toss tofu until it is evenly coated with cornstarch and arrange in one layer on the prepared baking sheet.
Bake tofu in the preheated oven for 10 minutes. Flip and continue baking until tofu is crisp, about 10 minutes more. Transfer tofu to a wire rack using a spatula; cool.
While tofu bakes, combine apricot preserves, vegetable broth, ginger paste, garlic, and red pepper in a small saucepan over low heat. Heat and stir sauce until warmed and well combined, about 5 minutes. Remove from heat. Serve tofu with warm dipping sauce.