Place the cream cheese in a medium bowl and let it sit at room temperature until slightly softened, about 10–15 minutes if needed.
While the cream cheese softens, wash the celery stalks under cold water, trim off the leafy tops and base, and cut each stalk into 3–4 inch sticks. Pat them dry well with paper towels so the filling adheres.
Add the sour cream, lemon juice, kosher salt, black pepper, and minced chives to the softened cream cheese.
Beat the mixture with a wooden spoon or small whisk until smooth, creamy, and evenly combined, 1–2 minutes. Taste and adjust seasoning with a pinch more salt or pepper if needed.
Using a small spoon or a piping bag fitted with a plain tip, fill the hollow side of each celery stick with the cream cheese mixture, mounding it slightly above the edges.
Arrange the filled celery sticks on a serving platter. Lightly sprinkle the tops with a little extra minced chives and a dusting of paprika if using.
Serve the celery and cream cheese immediately, or cover and chill for up to 4 hours. If chilled, let sit at room temperature for about 10 minutes before serving for the best texture.