Heat butter and oil in a large skillet over medium heat. When hot, add chopped shallots, then season with kosher salt. Cook until translucent, then stir in chopped garlic.
Increase the heat to medium-high, then add orzo, tomatoes, and Italian seasoning. Cook, stirring constantly, then pour in white wine. Cook, then add chicken stock. Bring to a boil. Cover tightly with a lid, then simmer until the orzo is al dente.
Stir in fresh baby spinach, goat cheese, lemon zest, and fresh lemon juice. Stir in pulled rotisserie chicken, grated Parmesan, and thinly sliced basil. Add lots of freshly cracked black pepper to taste. Stir until well mixed.
Taste and adjust seasoning, then transfer to individual bowls. Top with additional remaining goat cheese if wanted, then drizzle with extra virgin olive oil. Finish with flaky sea salt, then enjoy!