To roast the garlic, cut the ends of the garlic bulbs off to expose the garlic cloves, drizzle with EVOO and season with salt and pepper. Place garlic on foil and top with rosemary, wrap tightly in a foil pouch and roast until tender and light golden in color, 45-50 minutes. Remove and let cool; then squeeze garlic cloves from their skins and mash into a paste.
Place potatoes in a pot, cover with water and bring to boil. Season with salt and cook to tender, 12-15 minutes. Add a little of the starchy cooking water to beaten egg to temper it then drain potatoes and return them to the hot pot. Mash potatoes with tempered egg, roasted garlic, butter, milk and cheese; season with salt and pepper to taste.
For the casserole, heat a few inches of water to a boil, salt it and boil broccolini 5-6 minutes. Cold shock and drain.
Heat a large skillet over medium heat, add butter and melt it. Add shallots or onions, thyme, salt and pepper, and cook until tender, 12-15 minutes, stirring occasionally. Sprinkle in flour and stir 1 minute then whisk in stock and thicken. Stir in cream and heat through. Stir in Dijon and remove from heat. Combine sauce with broccolini and 4 cups of the poached chicken meat and transfer to a baking dish. Top with potatoes. (If not serving right away, let cool and refrigerate. Bring to room temp before baking.)
Place potatoes in a pastry bag or large plastic bag. Cut the corner of bag to form make-shift pastry bag to pipe potatoes across top of casserole or into mounds on a baking sheet for individual potato side-dish swirls. Bake at 400°F until golden, approximately 45 minutes.
Bake until hot through and golden, and serve.