Boil, or better steam, the potatoes and green beans for about 5 minutes, then add the zucchini and continue until all the vegetables are fully tender. Drain.
Pass the vegetables through a food mill, using the medium dish. Mix well and place the resulting purée in a non-stick skillet over gentle heat, stirring constantly, until any excess liquid has evaporated. You'll know it's done when the purée thickens so it comes away easily from the bottom of the pan. Transfer the purée to a mixing bowl and let it cool completely.
Add the rest of the filling ingredients to the bowl and mix until the mixture is smooth and homogeneous. The purée should be rather thick. If need be, mix in some more grated Parmesan to tighten it up. Taste and adjust for seasoning; the purée should be quite savory.
Now for the slightly tricky part: Open each zucchini blossom and gingerly remove the pistil inside, taking care not to bruise the petals or opening them too much.
Fill the center of each zucchini blossom with some of the purée. This is best done with a pastry bag if you have one. If you don't, you can improvise by snipping off one corner of a ZipLock or other plastic bag. (In a pinch, you can spoon the filling into the blossom, but it's rather tedious and delicate work.) Fill them well, but leave some room at the tip.
As you fill each blossom, twist the end to close it up, then place them in a pre-oiled baking dish. Drizzle everything with olive oil.
Bake in a hot (200C/400F) oven for about 20-30 minutes, until the zucchini blossoms are heated through and have browned on top.
Serve warm or at room temperature.