Make chickpea cake: Heat 1 tbsp. oil in a large, wide pot over medium heat. Cook onion, stirring often, until translucent, about 7 minutes. Add broth and heat until simmering. Sprinkle in chickpea flour and yeast, whisking until smooth.
Transfer mixture to a food processor. Purée until as smooth as baby food. Stir in thyme, fava leaves, and salt. Pour into a greased 9- by 13-in. baking dish. Lay a sheet of plastic wrap on top and use your hand to level mixture. Chill until cold, at least 1 1/2 hours and up to 2 days.
Invert cake onto a cutting board. Cut into 12 squares, then cut each square again diagonally to make 24 triangles.
Heat 1 tbsp. oil in a large nonstick frying pan over medium-high heat. Working in batches, brown triangles, turning once, 8 minutes total; add oil as needed. Set on 6 plates.
Make salad: Toss arugula, fennel fronds, parsley, and half the vinaigrette in a bowl. Spoon salad over chickpea triangles and serve remaining dressing on the side.
*Find chickpea flour and nutritional yeast at natural-food stores or well-stocked grocery stores. Find fava leaves at farmers' markets; if using spinach leaves instead, increase chickpea flour to 2 cups.
Note: Nutritional analysis is per serving with 3 tbsp. dressing. Total time does not include chilling time.