Step 1
Heat ¼ cup extra-virgin olive oil in a small saucepan over medium-high. Add 2 tsp. fennel seeds and 2 tsp. mustard seeds; cook, swirling pan, until fragrant and oil around seeds is sizzling, about 1 minute. Transfer spiced oil to a medium heatproof bowl.
Step 2
Place ¼ cup all-purpose flour on a large plate. Working one at a time, place 4 skinless, boneless chicken breasts, patted dry, between 2 large sheets of wax paper or plastic wrap. Using a meat mallet or rolling pin, pound on a surface to ¼” thick. Season cutlets all over with kosher salt and freshly ground pepper, then press into flour to lightly coat on both sides.
Step 3
Heat 3 Tbsp. extra-virgin olive oil in a large skillet over medium-high. Working in 2 batches and adding more oil to pan between batches if needed, cook cutlets until golden underneath, about 3 minutes. Turn over and cook on other side until cooked through, 1–2 minutes. Transfer to a plate.
Step 4
Heat remaining 3 Tbsp. extra-virgin olive oil in same skillet over medium-high. Arrange 1½ lb. cherry tomatoes in a single layer and cook, undisturbed, until blistered underneath, 1–2 minutes. Season with a big pinch of salt and cook, shaking pan occasionally, until softened, 1–2 minutes. Using a potato masher or spoon, carefully press down on tomatoes to burst and release seeds. Cook, stirring, 30 seconds, then transfer tomato sauce to bowl with spiced oil. Stir in 3 garlic cloves, finely grated, one 2″-piece ginger, peeled, finely grated, and 1 Tbsp. red wine vinegar; season with more salt if needed.
Step 5
Place a chicken cutlet on each plate. Spoon some tomato sauce over and top with cilantro leaves with tender stems.