Heat oven to 350 F.
To make the soup: Melt the butter in a Dutch oven over medium-high heat. Add the onions and sauté until soft, about 3 minutes.
Stir in the garlic and cook until fragrant, 30 seconds.
Sprinkle the flour over the onions and garlic and whisk to form a paste. Cook, whisking, for 1 minute.
Slowly pour in the half & half and chicken stock, while whisking, to incorporate the flour mixture.
Whisk in the Dijon mustard.
Bring the mixture to a simmer, stir in the broccoli and carrots, and return to a simmer. Cook over low heat for 20 to 25 minutes, until the soup is thick and the vegetables are tender.
Meanwhile, to make the breadsticks: Cut the focaccia in half and then slice each half into ½-inch thick batons and place on a rimmed baking sheet, leaving space in between the breadsticks.
Drizzle the olive oil over the breadsticks and season with salt and pepper. Sprinkle the Parmesan over the breadsticks.
Bake for 7 to 10 minutes, until lightly brown. Cool on the baking sheet.
To finish the soup, sprinkle in the cheese, about ½ cup at a time, whisking and melting the cheese between each addition. Repeat until all the cheese is added.
Stir in the nutmeg and season the soup with salt, pepper and Tabasco, to taste.
Ladle the soup into bowls, garnish with more cheddar and parsley and serve with breadsticks.