Heat oven to 425°F. Bake dinner rolls as directed on bag. Cool completely, about 10 minutes.
With serrated knife, cut each roll in half crosswise. Spread cream cheese evenly on cut sides of roll halves.
Place 2 cheese slice quarters on bottom half of each roll. Top each with 2 turkey slices, 1 tablespoon chutney and one-eighth of the spinach; cover with top halves of rolls.