Bring the filtered water to a boil in a small stainless saucepan, turn off the heat, and stir in the honey until dissolved. Add the rosemary, cover, and steep off heat for 1 to 2 hours.
Remove twiggy rosemary stems. If using the optional qualifying Dijon, stir it into the syrup until fully mixed.
Preheat the oven to 350°F. Set the whole trimmed cauliflower in a stainless steamer basket over simmering water and steam about 40 minutes, until a fork slides through the thickest stem without resistance.
Put the chicken thighs in the covered Artichaut baking dish. Spoon the syrup over them, drizzle with the melted chosen fat, season with salt and pepper, and cover.
Bake covered about 45 minutes, until the thickest chicken reaches at least 165°F. Remove the chicken and measure 1/2 to 3/4 cup pan drippings. Lower the oven to 250°F and keep the chicken warm.
Break the tender cauliflower into chunks in a saucepan. Blend one-third to one-half of it with the 1/2 teaspoon salt and one-third to one-half of the measured pan drippings until creamy.
Fold the puree through the remaining cauliflower chunks. Smash some chunks with a fork and warm over the lowest heat until creamy and chunky with a light pan-juice sheen.
Serve the chicken over the smashed cauliflower and spoon any remaining measured pan juices over both.