Melt the butter in a 4 quarter soup pot or Dutch oven set over medium heat. Add the onions and saute until soft (about 5 minutes). Stir in the garlic, along with a big pinch of salt and black pepper.

Sprinkle the flour over the aromatics and stir to create a paste, letting the raw flour cook off for a minute or two.

Whisk in the milk 1/3 cup at a time, incorporating completely before adding the next pour. You're making the basis for the cheese sauce here. The bechamel will be very thick and should not have any lumps of flour.

Next, add the remaining liquids and the mustard, mixing thoroughly (a whisk is helpful here), and raise the heat to medium-high to bring the soup to a gentle boil. Note that the broth will be very thin now. No worries, it will thicken as we go!

Keep in mind that with dairy in the pot, you'll want to stir frequently throughout the rest of the recipe to prevent the dairy from scorching the bottom of the pan.

Add the shredded carrots and broccoli. Adjust the heat so that the soup maintains an active simmer (not an aggressive boil) and cook for 10 minutes. Don't forget to stir!

While the broccoli cooks, bring a separate medium pot of water to boil and cook the gnocchi per package directions (potato gnocchi usually cooks in about 3 minutes; they're ready when they've risen to the surface and are floating). Drain and set aside.

Fish out one piece of broccoli with a spoon and some broth and taste, to make sure the broccoli is tender and the broth is flavorful. Adjust with more salt and pepper as needed.

Reduce heat to low and let the soup rest for several minutes, until the bubbling stops.

Begin adding the cheese, one handful at a time, stirring constantly. The soup will become rich and creamy.

Finish by adding the cooked gnocchi to the soup and gently stirring. Give the soup one more taste, and adjust with salt and pepper as needed.