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Place chickpeas on a paper-towel-lined plate to allow any excess liquid to be absorbed.
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Place flour in a deep bowl; add chickpeas and toss until completely coated with flour.
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Heat 3 tablespoons olive oil in a large saucepan over high heat. When oil begins to smoke, add chickpeas, shaking pot to coat chickpeas with oil; chickpeas will begin to snap and pop. Keep shaking or stirring pan occasionally, until chickpeas are golden brown all over, about 4 minutes. Season with 1/4 teaspoon salt and pepper; using a slotted spoon, transfer chickpeas to a plate and set aside.
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Place pot over medium-high heat and add remaining tablespoon oil. Add garlic and cook, stirring constantly, for 30 seconds.
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Add red pepper and spinach to pot and cook until spinach begins to wilt, about 1 minute; season with remaining 1/4 teaspoon salt and pepper. Continue cooking until spinach has completely wilted, about 30 seconds more.
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Transfer spinach to a serving plate and top with chickpeas; serve immediately.