Preheat the oven to 400°F.
In a large bowl, combine cubed butternut squash, one tablespoon of olive oil, sliced shallots, thyme, salt, and pepper. Toss to coat.
Line one third of a sheet pan with parchment. Spread the squash mixture on the remaining two thirds. Roast for 10 minutes.
While the squash is roasting, toss bread cubes in the remaining olive oil. Brush both sides of the chicken with oil and season with salt and pepper.
After 10 minutes, remove the pan. Add the bread cubes and garlic cloves around the squash. Place chicken breasts on the parchment-lined side. Return to the oven and roast for 25 more minutes.
While it cooks, whisk together the vinaigrette ingredients in a bowl or shake in a jar until combined.
Once roasted, slice or shred the chicken. Squeeze a garlic clove into each bowl. Top with vinaigrette and parmesan. Toss gently and serve.