Melt butter in a large Dutch oven over medium-low heat. Add onion, garlic, salt and black pepper. Stir to combine, cover and cook until onions soften, about 8 to 10 minutes.

Slice off and discard rough ends of broccoli stalks. Peel tough outer skin with a vegetable peeler and discard. Slice stalks into 1 to 2-inch pieces.

Cut broccoli tops close to the base of the florets and reserve florets.

Add stalk pieces to the pot. Pour in 6 cups of water and remaining salt. Raise heat to high and bring to a boil. Reduce to medium-low and cover. Simmer until stalks are tender, about 20-25 minutes.

Add broccoli florets to the pot, cover, and cook until bright green and easily pierced, about 4 to 7 minutes. Remove from heat and add remaining tablespoon of butter.

Carefully transfer several cups of the mixture to a blender and blend until smooth. Return the purée to the pot.

Add lemon juice and taste. Adjust seasoning with more salt, lemon juice, or black pepper.

Serve warm, garnished with a drizzle of cream or sprinkle of chives if desired.