Set up your grill for indirect heat. If using a charcoal grill, build the coals on one side only. If using a gas grill, heat one side only. Allow the temperature to reach 350 degrees F.
Place a heavy-bottomed saucepan over the direct-heat side of the grill. Add the butter and allow it to melt. Add the onions with a pinch of salt and cook until beginning to soften, 2 to 3 minutes. Add the peppers and cook until slightly softened, 2 more minutes. Add the flour and stir to coat the vegetables. Stir in a third of the milk at a time, smoothing out any lumps, and bring to a simmer. Simmer for 30 minutes, allowing the raw flour flavor to cook out.
Whisk in the Cheddar and cream cheese and bring back to a simmer. Season with salt and pepper, then stir in the ham and nutmeg. Crack the eggs into the sauce, leaving spacing in between them. Move the pan to the indirect heat side of the grill and cover the grill with a lid. Cook until the eggs are set, 10 to 12 minutes. Sprinkle all over with the chili flakes and serve.