Sausage With Charred Cabbage and Giardiniera

Name: Sausage With Charred Cabbage and Giardiniera
Ingredients: 2 Tbsp. plus ¼ cup extra-virgin olive oil; 1 lb. sweet Italian sausages (4–5); 2 lb. green cabbage (about 1 medium), cored, cut into 1 pieces; 2 garlic cloves, finely chopped; 2 tsp. sugar; 2 tsp. whole-grain mustard; 1 12-oz. jar vinegar-based giardiniera (Italian-style), 1/2 cup brine reserved; 1 tsp. Diamond Crystal or ½ tsp. Morton kosher salt, plus more; ½ tsp. freshly ground pepper, plus more; ½ cup torn basil leaves, plus more whole for serving