Heat the butter in a pot.
Add the chopped onion, rosemary and thyme, and cook over medium-low heat, stirring often, until the onion bits are translucent and soft. Add the minced garlic and cook for another half a minute.
Pour in the broth.
Add the peas and potatoes.
Season with salt and pepper.
Bring to the boil, lower the heat and cover. Simmer until the potatoes and peas and quite mushy. Taste every 10 minutes and adjust the seasonings, as needed.
If you have an immersion blender, puree the mixture right there in the pot. Otherwise, pour the contents of the pot into the blender and puree. Add the mint leaves and pulse a few times.
Taste one last time; add more salt and pepper, if necessary.
Ladle the soup into bowls. Top with shaved cheddar and whole mint leaves. Serve at once.