Season and sear shrimp: Pat raw shrimp thoroughly dry with paper towels. Season with a pinch of salt and black pepper. Heat 1 tablespoon of olive oil and 1 tablespoon of butter in a large skillet (cast iron preferred) over medium-high heat. Add shrimp in a single layer and sear for 1 to 2 minutes per side until pink, opaque, and lightly browned. Transfer immediately to a warm plate and set aside.
Sauté aromatics & blister tomatoes: In the same skillet, melt the remaining 2 tablespoons of butter with 1 tablespoon of olive oil over medium heat. Add the minced garlic and red pepper flakes; sauté for 30 seconds until fragrant. Add the halved cherry tomatoes and sun-dried tomatoes; cook for 2 to 3 minutes until cherry tomatoes begin to soften and burst their savory juices.
Build velvety cream sauce: Pour in the chicken broth, using a wooden spoon to deglaze and scrape up any caramelized fond from the bottom of the pan. Bring to a gentle simmer for 1 minute, then reduce heat to medium-low and stir in the heavy cream. Simmer gently for 2 to 3 minutes until the sauce slightly thickens into a glossy cream reduction.
Fold in spinach and parmesan: Add the fresh baby spinach and stir for 1 minute just until wilted into the sauce. Gradually stir in the freshly grated Parmigiano-Reggiano until completely melted, smooth, and velvety.
Return shrimp & finish: Return the seared shrimp and any accumulated resting juices back to the skillet. Gently toss to coat in the bubbling sauce and simmer for 1 minute to reheat through. Squeeze fresh lemon juice over top and garnish lavishly with fresh basil.
Serve immediately: Serve hot directly from the skillet with toasted garlic bread for dipping, or spoon generously over al dente fettuccine pasta or creamy parmesan polenta.