Prep the broccoli and cauliflower ahead of time and chill the vegetables in the fridge.
Fill a stockpot with water and bring to a boil over med-high heat. Blanch the broccoli and cauliflower for three minutes, then drain in a colander and rinse thoroughly with cold water.
Place a skillet over med-high heat, add two tablespoons of EVOO, when the oil is hot add the onions and sauté until the onions become soft, add the minced garlic and prosciutto, and continue to cook for another two minutes. Using a slotted spoon remove the onion, garlic, and prosciutto mixture and place it into a small bowl. Set aside.
Wipe the skillet clean. Add butter to the skillet, when butter is melted whisk in the all-purpose flour and whisk into the roux is a pale blond color.
Add 2 1/2 cups of heavy cream and continue to whisk until the mixture begins to thicken.
Whisk in the smoked paprika.
Fold in the grated Gruyère (or Parmigiano Reggiano). Add a pinch of salt, pepper, and freshly chopped rosemary. Continue to whisk until the sauce coats the back of a spoon. Remove from heat.
Brush a baking dish with olive oil.
Arrange the broccoli and cauliflower in a baking dish, pour the cheese sauce over the vegetables, add the onion, garlic, and prosciutto mixture. Add an additional 1/2 cup of heavy cream.
Bake in a 375-degree oven for 45 minutes.
Remove from the oven and serve warm.
