Preheat oven to 350°F / 175°C and line a baking pan with parchment paper.
In a large skillet on medium flame, heat extra virgin olive oil until shimmering. Add to it the thinly sliced onions, salt, and sugar, and cook down until golden brown, about 20 minutes, stirring occasionally.
While the onions are cooking, start a large pot of water with a palmful of kosher salt, about 1-2 tbsp and bring to a boil. Add orecchiette and cook until tender, about 12-15 minutes. Reserve a half cup of the pasta water and drain the rest. Return orecchiette to the pot.
Move the onions to the side of the pan. Melt 1 tbsp of butter in the pan, and when hot, add the garlic to it. Stir until fragrant, about 20 seconds. Then add the garlic to the onions to combine. Add all of the broccolini to the pan, stirring to combine. Add pasta water to the pan. Cook another 20 minutes, stirring occasionally.
Arrange the prosciutto slices on the baking pan. Bake in the oven for 12 minutes, until lightly crisp.
Add the Parmesan and broccolini mix to the orecchiette. Break apart pieces of the prosciutto and add it to the orecchiette, too, stirring it all to combine. Adjust for salt, if needed.
Serve topped with crushed croutons and more Parmesan, if desired.