Line a sheet tray with paper towels and arrange eggplant on paper towels in a single layer; season with 1 tsp. salt. Place another layer of paper towels on top. Let sit 10 minutes. Gently press down on paper towels to absorb excess moisture.

In a large stainless steel skillet over medium heat, heat 2 Tbsp. oil. Add half of eggplant; season with salt. Cook, tossing occasionally, until well browned, about 10 minutes. Transfer eggplant to a plate.

In same skillet over medium heat, heat 2 Tbsp. oil and repeat with remaining eggplant.

In same skillet over medium-high heat, heat remaining 1 Tbsp. oil. Cook onion, stirring occasionally, until just starting to brown, about 10 minutes. Add garlic, thyme, red pepper flakes, and a pinch of salt. Cook, stirring frequently, until garlic is fragrant and starting to brown, about 3 minutes. Stir in tomatoes, broth, and Italian seasoning and bring to a boil over high heat. Stir in orzo and return to a boil. Reduce heat to medium-low and bring to a simmer. Cook, stirring frequently to keep orzo from sticking, until orzo is al dente and liquid is thick and saucy, 10 to 12 minutes.

Place a rack in closest position to broiler; turn on broiler. Add spinach and Parmesan to orzo mixture and stir until spinach is wilted. Return eggplant to skillet and stir until distributed. Top with mozzarella.

Broil eggplant Parm, watching closely, until mozzarella is melted and starting to brown, about 5 minutes.

Top with basil and more Parmesan.