Shred the cheese and set it aside along with the Velveeta to allow it to come closer to room temperature. It will melt creamier this way.
Add olive oil to a large soup pot over medium heat. Add the diced onions and saute for 5 minutes. Remove the onions and set aside.
Melt the butter in the same pot over medium heat. Whisk in the flour and cook for 1 minute, until you can no longer smell flour. Use a silicone spatula to stir as it cooks.
Slowly stir in the chicken broth and whisk to combine until no lumps remain.
Add the onions back to the pot.
Add the half and half, paprika, mustard powder, garlic powder, and pepper. Bring to a boil, reduce to a simmer. Optional: Combine 3 TBS corn starch + 3 TBS cold water and whisk it in when the boil is reached. This will thicken it further.
Add the broccoli and carrots. Simmer for 10-15 minutes, until the broccoli is tender.
Optional: Remove 1-2 cups of broccoli/carrots and use an immersion blender to blend the soup for a smoother and creamier consistency. Add the remaining broccoli and carrots back.
Remove from heat and let the base cool down for 1-2 minutes. Gradually add the shredded cheese and Velveeta cheese, stirring as you do so.
Taste, and season with additional salt if desired. Serve!