In a large pot of lightly salted water, cook the broccoli until very soft, drain and set aside.
In a large saucepan over medium-high heat, melt 4 tbs butter. Add the onion, carrots, and celery and cook until the carrots are very soft, about 8 to 10 minutes. Add the garlic, cook for about 30 seconds, remove from heat.
In a large food processor or blender, add 1 cup beer, cornstarch, both kids of cheese and as well as the carrot and celery mixture, process until very smooth, about 5 to 8 minutes. Return mixture to the pot along with the remaining beer.
In the same food processor (no need to clean between jobs) add the broccoli and the broth, process until very smooth. Add pureed broccoli, as well as the cream, to the soup.
Bring the soup to a low simmer, stirring occasionally until warmed and slightly thickened. Salt and pepper to taste.
In a skillet over medium-high heat, melt 2 tbs butter. Add the bread cubes and toss until browned. Remove from heat and immediately add in the pesto, toss to coat.
Serve the soup topped with croutons.