For the preserved lemon: Mix the salt with 1 cup sugar in a small bowl. Layer the lemon slices with the salt-sugar mixture in a nonreactive dish so that the lemon slices are completely buried. Leave the dish uncovered on the kitchen counter and let sit for 2 to 3 days.
Remove the lemon slices from the salt-sugar mixture and rinse well under cold water. Combine the remaining 1/2 cup sugar with 1/2 cup water in a saucepan and bring to a simmer. Cook until the sugar is completely dissolved, then add the lemon slices and simmer for 30 minutes. Let cool and store in a container with a tight-fitting lid in the refrigerator for up to 1 month.
For the broccolini: Preheat the oven to 400 degrees F.
Blanch the broccolini by filling a medium saucepan with water and bringing it to a boil. Add a generous handful of salt (it should taste like the sea). Prepare a large bowl of ice water. Add the broccolini to the boiling water and cook until just barely tender, about 2 minutes. Drain the broccolini and immediately plunge into the ice water to shock. Transfer the broccolini to paper towels and blot dry.
Add the blanched broccolini, garlic, 1 tablespoon olive oil and some salt and pepper to a cast-iron skillet and toss well. Place the skillet in the oven and roast until the broccolini stalks are soft and cooked through and the ends are brown and crispy, approximately 10 minutes. Meanwhile, mix the chile paste, 2 tablespoons chopped preserved lemon and 1 tablespoon olive oil in a small bowl and season to taste with salt and pepper.
Toss the roasted broccolini with the preserved lemon mixture and serve.