In a medium heatproof bowl, season broccoli with ¼ tsp. salt. Pour 1 Tbsp. water over broccoli, then cover and microwave 1 minute. Stir and continue to microwave in 30-second increments until broccoli is bright green and fork-tender, 30 seconds to 1 minute more. Uncover and set aside.

In a large skillet over medium heat, melt 1 Tbsp. butter. Crush crackers into skillet (a mix of larger and smaller crumbs is fine) and cook, stirring constantly, until golden brown and toasty smelling, 1 to 2 minutes. Transfer to a small bowl. Wipe out skillet.

In same skillet over medium heat, melt remaining 4 Tbsp. butter. Add onion; season with ½ tsp. salt. Cook, stirring occasionally, until onion is tender and sweet to the taste, 7 to 8 minutes. Add garlic and cook, stirring, until fragrant, about 1 minute more. Add flour and cook, stirring with a spatula, until a bubbling paste forms on bottom of skillet, about 1 minute more.

Switch to a whisk and slowly whisk in milk and remaining ½ cup water until incorporated. Bring to a simmer over medium-high heat and cook, stirring occasionally, until thickened, 1 to 2 minutes. Add chickpeas and baking soda; cook, stirring, until heated through, about 1 minute. Fold in broccoli.

Reduce heat to low and fold in cheddar until melted and smooth; taste and season with remaining ½ tsp. salt, if needed.

Top with buttery crackers and serve immediately.