Combine 2 cups of the chicken stock and one-third of the potatoes in a small saucepan. Bring to a boil, and then reduce to a simmer until fork-tender, 15 to 18 minutes. Stir in the mustard to combine, and then blend to a smooth puree using an immersion or regular blender.
Meanwhile, cook the bacon in a large Dutch oven or stock pot over medium heat until crispy and most of the fat has been rendered, 12 to 15 minutes, and then transfer to a paper-towel-lined plate to drain. Pour out the bacon fat, reserving 3 tablespoons in the pot.
Turn the heat up to medium-high and add the carrots. Sprinkle with salt and pepper and cook for 3 minutes, and then add the remaining potatoes, celery, bay leaves, leeks and thyme and season with salt and pepper. Cook until the leeks are soft, 4 minutes, and then pour in the beer and bring to a boil until the beer has cooked down to about 2 tablespoons, 2 to 4 minutes. Pour in the remaining chicken stock and the reserved potato puree. Stir to combine and bring the mixture to a boil. Reduce to a simmer and cook uncovered until the vegetables are just tender, 6 to 8 minutes. Season with salt and pepper.
To serve, ladle the soup into bowls. Crumble the reserved bacon on top and garnish with the chives.