Cook bacon over medium heat in a large pot until crispy, about 8 minutes. Transfer bacon to a plate. Measure out 3 tablespoons drippings and add back to pot (add butter or oil to reach 3 tablespoons if you don’t have enough).
Meanwhile, add leeks to a salad spinner or large bowl and fill with cold water. Shimmy leeks around in water using hands. Drain then repeat filling with water, shimmying and draining. Spin dry or pat dry using clean kitchen towels.
Add washed and dried leeks and cook over medium-high heat until starting to brown around edges, about 5 minutes.
Add garlic and 1 teaspoon salt and cook just until fragrant, 1 minute. Stir in broth, cauliflower, potatoes and whole-grain mustard. Bring to a boil then reduce to a simmer and cook until vegetables are tender, 20–25 minutes.
Transfer 4 cups of soup to a blender with milk, Dijon and ¼ teaspoon white pepper; blend until smooth
Pour back into pot and season with additional salt to taste, about 1 teaspoon.
Top servings with crispy bacon bits. Garnish with chives and lemon zest, if desired.