Discard the leaves from the broccolini and cut the stalks (up to the florets) into 1/2-inch pieces. Rinse in a colander and set aside.
Heat a large, heavy skillet over medium-high heat. Pat the fish dry and season with salt and pepper. Add 2 tbsp. EVOO to the skillet and swirl to coat. Add the fish and cook until nicely golden on the bottom, about 3 minutes. Flip the fish and cook until firm to the touch, about 3 minutes longer. Transfer to a plate and cover loosely with foil to keep warm.
Pour off and discard the cooking oil. Return the skillet to medium-high heat, add the garlic and 1 tbsp. EVOO and cook, stirring and scraping up the browned bits, until golden, 1 to 2 minutes. Add the wet broccolini, season lightly with salt and pepper and cook for 1 minute. Add the tomatoes, beans and 1 cup water and cook, stirring occasionally, until the broccolini is tender and the tomatoes are saucy, about 5 minutes. Stir in 2 tbsp. EVOO and season with salt and pepper.
Spoon the tomato-broccolini mixture onto 4 plates or shallow bowls. Top each portion with a piece of fish and drizzle with a little more EVOO.