Preheat oven 375°F.
In a skillet, add 1 tablespoon of canola oil over high heat. Add chicken breasts to skillet and cook until the chicken is longer pink, about 5-7 minutes. Remove chicken and allow to cool. Using the same skillet, add the bacon and cook until brown and crispy, about 3-5 minutes. Remove the bacon and set aside. Add butter to the bacon fat. Add onions, carrots, celery and potatoes and sauté for about 5 minutes, stirring occasionally. Add the minced garlic and stir for 1 more minute until fragrant.
Add flour to the vegetable mixture and stir for 1-2 minutes until toasty. Add chicken stock, bouillon cube salt and pepper and whisk until smooth and thick, about 1-2 minutes more. Reduce heat to a simmer.
After about 10 minutes check the potatoes to see if they are done. If so, stir in the Broccolini and herbs.
Divide mixture among four oven-proof soup bowls. Cut four circles out of the puff pastry, about ½-inch larger than the diameter of the bowl.
Whisk one egg and brush the outside edge of each bowl with the egg wash. Lay a circle of the puff pastry on top, and cut a small vent in the center. Brush the puff pastry with more egg wash. Repeat with the rest of the bowls.
Bake in oven until the top is golden brown, about 10-15 minutes.
Allow to cool for 7-8 minutes before serving.