Place a medium skillet over medium-high heat. Add 1 tablespoon of olive oil and the sausage to the skillet and cook, stirring often and breaking the sausage apart with the back of a wooden spoon, until the sausage is cooked and beginning to brown, about 4 minutes. Add the broccolini and salt and continue to cook until the broccolini is tender, an additional 6 minutes. Allow the mixture to cool.
In a medium bowl, mix together the sausage mixture and provolone until combined. Place 4 slices of bread on a cutting board and divide the mixture evenly among the slices. Top with the remaining slices of bread.
Heat a large skillet or griddle over medium heat. Brush with 1 1/2 tablespoons of the remaining oil and place 2 sandwiches in the skillet. Cook until lightly browned and crispy on the bottom, about 3 minutes. Flip the sandwiches and cook until crispy on the bottom and the cheese is melted, an additional 3 minutes. Continue with the remaining oil and sandwiches. Cut into squares or triangles and serve.