1. Boil broccolini in a pot of salted water until it's bright green, 2 to 3 minutes. Drain, rinse with cold water, and drain well. Set aside.
2. Cut tofu into very thin (1/8 in. thick) slices and put in a bowl. Stir together 1 tsp. sugar, 1/4 tsp. pepper, the vinegar, 1 1/2 tbsp. soy sauce, and 2 tsp. sesame oil in a small bowl. Put next to stove along with tofu, duck meat, and garlic.
3. Slice duck skin thinly and heat, along with fat, in a large wok or frying pan over medium-high heat. Cook fat and skin, stirring 2 to 3 minutes, until most of the fat has melted and skin is crisp. Transfer skin with a slotted spoon to paper towels. Pour off all but about 2 tbsp. fat from pan (if you don't have that much fat, add canola oil to make up the difference).
4. Add garlic to pan and let sizzle just until fragrant, about 15 seconds. Add tofu and a pinch each of salt and sugar. Cook 30 seconds, then add duck meat and stir-fry 30 seconds to heat through. Add broccolini and stir-fry 2 minutes to heat.
5. Stir seasoning sauce again, then add to wok. Cook until most of the liquid has been absorbed, about 1 minute. Season with extra salt or soy sauce if you like. Transfer to a plate or shallow bowl and sprinkle with remaining 1 tsp. sesame oil and white pepper to taste. Serve duck cracklings on the side so they stay crisp, for sprinkling on top.