Place cauliflower, onion, garlic, broth, 1/2 teaspoon salt and pepper in a 6-quart slow cooker. Cover and cook on low for 4 hours until cauliflower is very soft.
During the final 35 minutes, heat the oven to 425°F. Toss broccoli with olive oil and remaining salt on a rimmed sheet pan and roast for 20 to 25 minutes, turning once, until tender with browned edges.
Blend the cauliflower broth in the cooker with an immersion blender until smooth. Chop the roasted broccoli into small spoon-size pieces.
Stir broccoli and evaporated milk into the cooker. Cover and cook on high for 20 to 30 minutes until steaming, without leaving the broccoli to cook for another full hour.
Switch off the heat and stir in white cheddar in 4 additions, then Dijon, until smooth. Divide among 6 bowls.