Grate the block of sharp yellow cheddar cheese by hand using a box grater, then set aside at room temperature to warm slightly for smoother melting.

Melt the unsalted butter in a heavy bottomed Dutch oven over medium heat until the foaming stops, indicating pure butterfat is ready.

Add the finely minced yellow onion into the melted butter and sauté for three to four minutes until translucent and aromatic without browning.

Whisk in the all purpose flour steadily to create a smooth golden roux paste structure, cooking for three to four minutes while stirring constantly.

Pour in the low sodium chicken broth slowly in three stages while whisking constantly to prevent lumps, allowing each addition to incorporate smoothly.

Add the half and half gradually in the same manner, whisking until the liquid base is completely smooth without visible flour lumps.

Bring the mixture to a gentle simmer over medium heat with small bubbles breaking around the edges, then add the chopped broccoli florets and julienned carrots.

Reduce heat to low and simmer gently for twelve to fifteen minutes, stirring occasionally, until the broccoli reaches crisp-tender texture while maintaining vibrant color.

Stir in the ground mustard powder, sea salt, and cracked black pepper, then taste and adjust seasoning as needed.

Turn off the heat completely and remove the Dutch oven from the burner entirely to allow the liquid to stop simmering.

Fold the grated cheddar cheese into the hot soup one handful at a time, stirring gently after each addition until completely melted and smooth before adding more.

Let the soup rest off heat for two to three minutes to allow the emulsion to stabilize fully before ladling into serving bowls.