Put a large soup pot or Dutch oven on the stove over medium heat.

Add the butter to the pot and let it melt.

Add the chopped onion and chopped carrot to the pot. Cook, stirring often, for 5 minutes, until the onion looks soft and the carrot starts to lose its raw crunch.

While the vegetables cook, put the cornstarch in a small bowl. Pour in 1/2 cup of the milk and whisk until smooth.

Pour the cornstarch mixture into the pot with the onion and carrot. Stir for about 30 seconds so the vegetables are coated and there are no dry spots.

Pour the remaining 1 1/2 cups milk and all of the vegetable broth into the pot. Raise the heat to medium-high and bring the soup to a gentle boil.

Add the sliced broccoli stems to the pot. Cook for 2 minutes.

Add the broccoli florets to the pot. Cook for 3 minutes, until they turn bright green and start to soften.

Add the rice noodles to the pot. Use a spoon to press them down into the liquid as they soften.

Reduce the heat to maintain a steady simmer. Cook for 4 minutes, stirring once or twice so the noodles do not stick together.

Add the frozen peas to the pot. Simmer for 2 to 3 minutes, until the peas are hot and the noodles are tender but not mushy.

Turn the heat down to low. Let the soup stop bubbling for about 30 seconds so the cheese will melt smoothly.