Placed the cubed squash into a microwave-safe bowl and add the 2 tablespoons of water and ½ teaspoon salt. Cover the bowl with a damp paper towel and microwave on high for 5 minutes. Keep covered and set aside.
Drizzle 2 teaspoons of the olive oil into a large saute pan and bring to medium-high heat. Sprinkle the chicken breast cutlets with salt and black pepper on both sides and lay them in the skillet. Cook for 3-4 minutes until golden on the bottom. Flip the breasts and cook for another 3-5 minutes until the chicken is cooked through. Remove the chicken to a side plate and cover with aluminum foil to keep warm.
In a mixing bowl, add the chicken broth, maple syrup, both types of Dijon mustard, vinegar, cornstarch, and thyme and whisk together until combined. Set aside.
Add the remaining 2 teaspoons olive oil to the saute pan and bring back to medium-high heat. Add the cubed squash in a single layer and follow with the garlic. Cook for 1 minute until garlic is fragrant.
Pour in the sauce mixture from step 3 and stir. Once bubbling (should happen fairly quickly), reduce the heat to low and simmer for 4-5 minutes, stirring occasionally, until the squash is just tender and the sauce has thickened. Add the spinach leaves and stir for 1-2 minutes until wilted. Add the chicken back in and stir to coat in sauce. Cook for another minute or two to warm the chicken and serve warm.