In a 4–5 quart pot, melt the butter over medium heat until it foams lightly. Add the minced garlic and cook 30–60 seconds, stirring constantly, just until fragrant. Avoid browning the garlic - it can turn bitter.
Slowly pour in the heavy cream while stirring. Keep the heat at medium to medium-low and bring the cream to a gentle steam (small wisps, not a boil). Stir occasionally to prevent scorching on the bottom.
Whisk in the vegetable broth gradually so the liquids combine smoothly. Return the pot to a gentle simmer; you should see small, lazy bubbles around the edges.
Stir in the chopped broccoli, diced potato, and grated carrot. Adjust the heat to maintain a steady, gentle simmer and partially cover the pot.
Cook 20–25 minutes, stirring every few minutes, until the potatoes are easily pierced with a fork and the broccoli is tender.
Reduce the heat to low or remove the pot from the burner. Add the shredded cheddar in small handfuls, stirring after each addition until fully melted and silky. Keep the soup below a simmer at this stage to prevent the cheese from turning grainy.
Season with salt and freshly ground black pepper, tasting as you go. If the soup is thicker than you’d like, loosen with a splash of warm vegetable broth; if it’s thin, simmer a few extra minutes uncovered.
For an extra hearty presentation, serve the soup inside hollowed-out bread bowls - choose round loaves with a firm crust so they hold the creamy filling without getting soggy.
Garnish with a sprinkle of cheddar. Enjoy!