Note: if you have an Instant Pot this recipe is ideal for it. Saute the carrots, onions and celery in olive oil and then add the herbs, stock and wine (if using), then switch your Instant Pot to the "soup" function and let 'er rip. Otherwise, follow the directions in the next step to make the soup.
Make the soup: in a large stockpot, heat the olive oil over medium-low heat until the olive oil is just shimmering. Add the onions first, sauteing gently until the onions are starting to turn translucent. Add the celery and carrots and continue cooking for 6-8 minutes, until the veggies are soft. Add the garlic and continue to cook for 1 minute.
If you're using the wine, pour that in first and stir with the veggies, letting the wine cook down to about half its original volume (you can monitor it by taking note of the wine level when you first poured it into the pot, and watching it as wine evaporates and the level goes down). Add the stock, bay leaf, oregano and thyme and season with 1 teaspoon salt and 1 teaspoon pepper. Bring everything to a boil, then lower the heat to a simmer and let the soup simmer for 25-30 minutes.If you're using the broccolini for your soup, add the broccolini and let it simmer for the last 5-10 minutes of your soup simmering time.
Make the gnocchi: While the soup simmers, wash and peel the potatoes, then chop into 1-inch (3cm) chunks. Put the chunks into a medium-sized pot and add water to cover the potatoes by about an inch. Salt the water generously, cover with a tight-fitting lid and bring to a boil over high heat. Lower the heat to a simmer and cook the potatoes until they're good and tender, about 10-12 minutes.
Drain the potatoes in a colander and set the colander over the hot pot you just used. Return the pot to the hot stove (even though it's off) and let the potatoes steam dry in the colander for 2-3 minutes. Remove the pan from the heat, pour the potatoes into the pan and mash well, seasoning with salt and pepper. Fill the pan again with cold water, salt the water, and bring the pot to a boil.
While you're waiting for the water to boil, flour a clean work surface and pour the mashed potato on to it. Sprinkle the flour over the potato and scrunch everything together to make it stick. Roll out the dough into a long snake, and then, using a sharp knife, cut the snake into 1.5 inch segments. Lightly roll a fork over each segment to make a little grooved pattern on top. Assemble your gnocchi onto a parchment paper lined baking sheet.
Once the water is boiling in your pot, add the gnocchi and let boil for one minute -- you'll see the gnocchi go from soft little pillows into tight little wads of potato. Fish them out with a strainer and divide them into 4 bowls for serving.
Ladle the chicken soup on top of the gnocchi and garnish with basil. Serve immediately.