Place milk, chicken broth, Vegeta, thyme, bay leaves, and pepper in a large saucepan over medium heat and bring to a low simmer. Add chicken and cover with a lid. Simmer on medium-low for 15 minutes without allowing it to boil to prevent curdling. Remove chicken, shred it, and keep the poaching liquid covered and reserved.

In a large, deep skillet over high heat, warm half the butter and olive oil. Add half the mushrooms, garlic, salt, and pepper, and sauté for 5 minutes until golden. Remove to a bowl and repeat with the remaining mushrooms.

Lower heat to medium and melt butter in the same skillet. Sauté garlic until fragrant and golden. Stir in the flour and cook while stirring for 1 minute. Gradually whisk in 2 cups of the reserved poaching liquid to form a thick paste. Continue to incorporate another cup, then the remainder, stirring until smooth. Stir over heat for 1–2 minutes until the sauce thickens to coat the back of a spoon. Remove from heat and stir in cheddar until melted. Adjust seasoning as needed. Reserve 2 1/2 cups (625ml) of sauce for topping, and mix shredded chicken and mushrooms into the remaining sauce.

Preheat oven to 180°C (350°F). Spread a small amount of sauce on the bottom of a 23 x 33 cm (9 x 13") baking dish. Layer with lasagna sheets, half the chicken mixture, another layer of pasta, the remaining chicken mixture, and a final layer of lasagna sheets. Pour over reserved white sauce and sprinkle with mozzarella. Cover loosely with foil or place an inverted baking tray on top, ensuring it doesn't touch the cheese.

Bake for 25 minutes, then uncover and bake an additional 10 minutes until the top is golden and bubbling. Let rest for 10 minutes before slicing. Garnish with parsley if desired.