Melt 3 tablespoons of fat, stir in flour and sweat for about 2 minutes. Peel and wash the potatoes, cut into cubes and cook in boiling salted water for about 15 minutes. Peel carrots and cucumbers.
Cut the cucumbers lengthwise in half and remove the seeds. Quarter the cucumber and cut into pieces. Cut carrots into slices. Boil eggs in boiling water for about 10 minutes until hard. Pour milk and stock into the roux while stirring.
Bring to the boil and simmer for about 3 minutes. Stir in mustard. Keep sauce warm. Melt 2 tablespoons of fat. Braise carrots for about 4 minutes, after 2 minutes add cucumber. Add béchamel, bring to the boil and simmer for about 3 minutes, stirring continuously.
Drain and fold in the potatoes. Wash chives, shake dry and cut into small rolls. Quench eggs cold, peel and halve. Arrange ragout with eggs and chives.