Preheat the oven to 400 degrees F (200 degrees C).

Place broccoli, baby potatoes, carrot, and onion in a large bowl. Pour in olive oil, mustard, Pecorino Romano cheese, thyme, oregano, salt, and pepper. Toss well to coat, then spread vegetables in an even layer on a sheet pan. Arrange sausages in the pan, pushing aside the vegetables so that the sausages touch the bottom of the pan.

Bake in the preheated oven, flipping sausages and vegetables halfway through until sausages are no longer pink in the center, 30 to 35 minutes. An instant-read thermometer inserted into the center should read 160 degrees F (70 degrees C).