Heat oil in deep skillet; soften onion, carrot, and celery over medium 8 to 10 minutes. Add chicken; cook 6 to 8 minutes, breaking finely, to 165°F.
Stir paste 1 minute; add wine and simmer 2 minutes. Add tomatoes, broth, milk, bay, sauce salt, and pepper.
Gently simmer partly covered 30 to 35 minutes, stirring occasionally, until thick but spoonable. Remove bay.
Near end, cook tagliatelle with tablespoon salt to 1 minute short of package time. Reserve 1 cup water; drain.
Toss pasta into sauce over low 1 to 2 minutes, gradually adding water if needed. Divide measured Parmesan over servings.