Heat oven to 450ºF (232ºC) with rack set in bottom and middle.
Wash and cut potatoes into ¼-inch thick wedges.
Mix together oil, dill, shallot, Dijon, 1 teaspoon salt, and red pepper flakes.
Coat a baking sheet with nonstick spray.
On prepared baking sheet, toss potato wedges with 2 tablespoons oil mixture and ½ teaspoon salt to coat; arrange in an even layer. Transfer to bottom rack in oven and roast 15 minutes.
Using a vegetable peeler, peel tough outer skin from broccolini stems.
On a second sheet pan, toss broccolini with 2 tablespoons oil mixture and ¼ teaspoon salt. Arrange in an even layer then sprinkle tomatoes all over sheet pan. Transfer to middle rack.
Remove potatoes from oven, gently flip wedges, then nestle chicken sausage around potatoes. Transfer sheet back to bottom rack.
Roast both sheet pans until potatoes and sausages are golden brown, broccolini is tender and some of the tomatoes have burst, about 20 minutes, flipping sausages and stirring potatoes halfway through.
Meanwhile, whisk vinegar into remaining oil mixture.
When ready to serve, dump broccolini and tomatoes onto sheet with potatoes and sausages. Drizzle with oil mixture and serve.