In a small saucepot, place potatoes and cover with cold water. Place over high heat and bring up to a bubble. Cook until tender, about 15 minutes.
In another small saucepot, add parsnips, celeriac and milk, and bring up to a bubble. Cook until tender, about 20 minutes.
When everybody is tender, scoop a mug of hot milk out of the parsnips and celeriac, then drain and smash potatoes, parsnips, and celeriac together along with nutmeg, butter, reserved hot milk, salt and ground black pepper.
While root veggies are working, season chicken breasts with salt, pepper and fresh thyme then wrap each breast in a piece of ham. Heat 1 turn of the pan of EVOO, about 1 tablespoon, in a large saut pan over medium-high heat. Cook ham-wrapped chicken breasts until crisp and cooked through, about 6 minutes per side, 12-15 minutes total.
While the chicken is cooking, prepare the gravy: Place a medium saut pan over medium-high heat with 1 tablespoon butter. Cook the shallots in the butter until tender, about 2 minutes. Add the flour to the pan and cook until combined, about 1 minute. Whisk in the stock, syrup and vinegar, and bring up to a bubble. Cook until thickened and reduced, 3-4 minutes.
To serve, place a chicken breast on each plate topped with the Maple Gravy and some Smashed Root Veggies alongside.