Beat the egg in a medium bowl and whisk in milk, pesto, and basil.
Add the flour, sugar, baking powder, and salt. Stir to make thick dough and set aside.
Soup
Heat oil in a large pot over medium heat. Add onion and sauté for 2 minutes, then add carrots and cook 5 more minutes. Stir in garlic and flour.
Add stock, herbs de Provence, salt, and pepper. Bring to a boil, add beans, then reduce heat to medium-low and simmer for five minutes then add the peas.
Finish
Drop tablespoons of dumpling mixture into the soup and cook for 6 minutes.
Flip dumplings and cook another 6 minutes, keeping the soup at a low simmer and covered.
Serve hot with extra pesto.