1. In a large nonstick skillet over medium heat, toast walnuts until lightly browned and fragrant, about 5 minutes. Remove from pan.
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2. Add oil to pan and heat over medium-low heat. Add shallots and cook, stirring often, to a deep golden brown, about 10 minutes. (Add a tablespoon or two of water if the mixture gets too dry.) Season with salt and pepper; set aside in the skillet.
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3. Meanwhile, steam broccoli until just tender, 3 to 5 minutes. Add broccoli and walnuts to the shallots in the skillet and toss to combine. Taste and adjust seasonings.
