Bring water with chicken to a boil in a wide 6-quart heavy pot, skimming froth. Add thyme sprigs, bay leaf, peppercorns, and 1 teaspoon salt, then reduce heat and simmer, uncovered, 10 minutes.
While chicken simmers, peel carrots and parsnips, then cut each carrot and parsnip crosswise into 3 equal lengths. Halve or quarter thicker pieces lengthwise so pieces are roughly the same size. Add to chicken along with onion, celery, and garlic and simmer, uncovered, until parsnips are tender, 10 to 15 minutes more. Transfer carrots and parsnips with a slotted spoon to a large bowl, then continue to simmer chicken until tender, 5 to 10 minutes. Transfer chicken with tongs to a bowl, then pour broth through a sieve into another large bowl. Let chicken and broth cool, uncovered, 15 minutes.
Chill broth, uncovered, until fat rises to top and solidifies, about 4 hours.
Meanwhile, tear chicken into bite-size pieces, discarding skin, bones, and remaining solids in sieve. Chill chicken with carrots and parsnips, covered, about 4 hours.
Remove and reserve 1/4 cup solidified fat from stock. (If you have less than 1/4 cup fat, add butter.) Melt fat in a wide 4-quart heavy saucepan over moderately low heat, then add flour and cook roux, stirring constantly, 3 minutes (do not brown). Stir in chicken broth. Bring to a boil, stirring constantly, then reduce heat slightly and simmer briskly, uncovered, stirring occasionally, until reduced to about 2 1/2 cups, 30 to 50 minutes. Add milk and cream, then simmer until reduced to about 3 1/2 cups, 10 to 15 minutes. Stir in chicken with vegetables, remaining 1/2 teaspoon salt (or to taste), and vinegar and sugar to taste and cook, stirring gently (avoid breaking up vegetables), until heated through. Stir in dill.
Cooks' Note: Stew, without dill, can be made 1 day ahead and cooled, uncovered, then chilled, covered. Reheat over moderately low heat, then add dill.

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Nutrition Facts

Per Serving Per 100 g

Amount Per 1 Serving
Calories	1128.59 Kcal (4725 kJ)
Calories from fat	909.95 Kcal
		% Daily Value*
Total Fat	101.11g	156%
Cholesterol	247.96mg	83%
Sodium	773.06mg	32%
Potassium	873.9mg	19%
Total Carbs	30.86g	10%
Sugars	10.04g	40%
Dietary Fiber	6.4g	26%
Protein	24.56g	49%
Vitamin C	19.6mg	33%
Vitamin A	0.7mg	23%
Iron	0.9mg	5%
Calcium	132.7mg	13%

Amount Per 100 g
Calories	254.52 Kcal (1066 kJ)
Calories from fat	205.21 Kcal
		% Daily Value*
Total Fat	22.8g	156%
Cholesterol	55.92mg	83%
Sodium	174.34mg	32%
Potassium	197.08mg	19%
Total Carbs	6.96g	10%
Sugars	2.27g	40%
Dietary Fiber	1.44g	26%
Protein	5.54g	49%
Vitamin C	4.4mg	33%
Vitamin A	0.2mg	23%
Iron	0.2mg	5%
Calcium	29.9mg	13%

* Percent Daily Values are based on a 2000 calorie diet. Your daily values may be higher or lower depending on your calorie needs.
Find out how many calories should you eat.

Tastes

salty
savory
bitter
sweet

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