Soak the wooden skewers in water for 10 minutes while you prepare the other ingredients. Thread the sausage pieces and pumpkin cubes onto the skewers, alternating them evenly.
Preheat the oven to 400 degrees F and line a baking sheet with parchment paper. Place the assembled skewers on the sheet in a single layer.
In a small bowl, stir together the maple syrup, olive oil, cinnamon, salt, and black pepper. Brush the glaze generously over all sides of the skewers.
Bake for 20 minutes, turning the skewers once halfway through. The pumpkin should be fork-tender and the edges of the sausage lightly browned.
Remove from the oven and let the skewers rest for 5 minutes. Garnish with fresh thyme sprigs before serving.