Place potatoes in a medium pot of lightly salted water, just enough to cover potatoes. Bring potatoes to a boil over medium-high heat, then reduce heat to medium and let potatoes simmer until just barely tender, about 5-8 minutes (a paring knife should go through a cube with just a bit of resistance) Drain potatoes and pat dry with paper towels.
Heat 1 tablespoon vegetable oil in a large skillet over medium-high heat. Add chicken sausage in a single layer and sear on both sides until golden brown. Remove from pan and set aside.
In same skillet, heat 1 tablespoon vegetable oil over medium-high heat. Add zucchini, summer squash, and red onion and saute 4-6 minutes until vegetables are tender and lightly browned. Add cherry tomatoes and garlic and cook 1 additional minute until fragrant. Remove from pan and set aside.
In same skillet, heat remaining 2 tablespoons vegetable oil over medium-high heat. Add pre-cooked potatoes in a single layer in pan. Let potatoes cook without stirring 3-5 minutes until crispy, then use spatula to flip to opposite side and cook an additional 3-5 minutes until crispy.
Add chicken sausage and sauteed vegetable mixture to potatoes and toss until evenly combined. Season hash with salt and pepper to taste and top individual servings with eggs, Parmesan, and fresh basil. Enjoy!