Sausage and bread are always a good move—but they're even better with garlicky broccolini.
In a deep skillet or Dutch oven, heat 2 tbsp. oil, two turns of the pan, over medium-high. Add the butter. When it foams, add the garlic and chile paste. Stir for 1 minute, then add the stock and broccolini. Cover and cook until the broccolini is crisp-tender, 5 to 6 minutes. Uncover and toss until the edges of the broccolini are crispy and the stock is completely absorbed, 1 to 2 minutes. Season with salt and pepper.
Meanwhile, using a sharp knife, pierce the casing of the sausages. (If using plant-based sausages, you may need to remove the casing.) In a large skillet, add 1/8 inch water and the remaining 2 tbsp. oil. Add the sausages and cook, turning occasionally, until browned and cooked through, about 15 minutes. Transfer to a cutting board. Let rest a few minutes.
Cut the sausages on an angle into 1-inch pieces. Arrange the sausages on top of the broccolini and drizzle with the lemon juice. Drizzle the rolls with EVOO, then build the sandwiches using the rolls, the sausages, and the broccolini.