Bring the stock to a simmer on medium heat. Add the minced garlic, the herbs/spices, and the potatoes. Cover and simmer about 15 minutes.
Add the broccoli and simmer uncovered for another 5 minutes or so. (If using fresh broccoli, about eight minutes).
Beat the eggs in a small bowl. Remove a ladleful of the soup and whisk it into the eggs to temper them. Remove the soup from the heat and whisk in the egg mixture. The eggs should cloud the soup but they shouldn't scramble.
Top each serving with the grated Parmesan.
You can make the soup in advance and add the eggs just before serving if desired.
Variations: substitute 1 cup small pasta for the potatoes; add a few garlic croutons to each bowl before serving; substitute other green vegetables, such as green beans, peas, kale.