To start, prep the ingredients before you start. Have everything next to the stove in arms reach. Roughly chop the broccoli florets, dice the onion and mince the garlic.
In a large pot or dutch oven on medium high heat add a little oil then the diced onions. Cook the diced onions until they become translucent then toss in the garlic. Cook them both until you see brown bits forming on each.
Lower the heat to medium and toss in the butter. Once the butter has melted add in the flour and mix everything together cooking out the flour taste. After about 30 seconds start to slowly pour in the milk until mixed well and creamy. Then, add in the chicken stock
Next, toss in the broccoli cuts and carrots. Season everything with the salt and pepper. You'll taste later so no need for perfection.
Now, cover the pot for 5 minutes to steam and soften the broccoli. Keep the heat medium to avoid burning at the bottom of the pot. Turn off the heat.
Then, add in freshly shredded cheese. Mix everything together and taste for seasoning before serving.