Heat the olive oil in very large heavy saucepan over moderately high heat until ripples appear on the pan bottom, about 1½ minutes. Add the onion and garlic and cook, stirring occasionally, until soft, about 5 minutes.
Add the potatoes, flour, salt, and pepper, stirring well to mix. Slowly add the chicken broth and evaporated milk, stirring constantly. Bring quickly to a boil while continuing to stir.
Reduce the heat to low, cover, and simmer until the potatoes are tender, about 20 minutes.
Add the broccoli, parsley, and basil and simmer uncovered, stirring now and then, until broccoli is crisp-tender, about 10 minutes.
Ladle into soup bowls and serve.
Notes
If you are preparing this recipe as gluten free, just be sure to use a brand of broth that is known to be GF, and use a thickener replacement for the all-purpose flour, such as potato or tapioca starch.