To prepare compote, combine first 7 ingredients in a medium saucepan. Bring to a boil. Cover, reduce heat, and simmer 20 minutes, stirring occasionally. Simmer, uncovered, until mixture reduces to about 1 cup (about 30 minutes). Remove from heat; cool.
Place fig mixture in a food processor; pulse 5 times or until finely chopped. Place in a small bowl; cover and chill.
To prepare tarts, preheat oven to 350°.
Arrange phyllo shells in a single layer on a jelly-roll pan. Combine cheeses in a medium bowl; beat with a mixer at medium speed until smooth. Add egg white; beat well. Add milk; beat well. Combine sugar, flour, and salt; add to cheese mixture, beating well.
Spoon about 1 1/2 teaspoons cheese mixture into each phyllo shell. Bake at 350° for 15 minutes or until lightly browned. Cool on a wire rack. Top each tart with about 1 1/2 teaspoons compote.