Cornmeal Souffle with Sage Browned Butter

Originally a failed recipe from Elizabeth Davis' book French Country Cooking, Gnocchi à la Romaine with sage brown butter sauce. I misread and used cornmeal instead of wheat semolina, therefore I feel like it has become it's own thing. A bit stodgy on its own, I've included a few suggestions for what could go well with it at the end of the recipe.

Timing: 

6 cups milk

1½ tsp salt

1 tsp pepper

¾ tsp nutmeg

3 Tbsp minced chives

12 oz (340g) cornmeal, fine or medium grind

⅓ cup and 1 rounded Tbsp (3oz/85g) butter + 2 oz butter + extra

⅓ cup and 1 rounded Tbsp (3oz/85g) + ⅓ cup and 1 rounded Tbsp grated Asiago or Gruyère

3 eggs, whisked

8 fresh sage leaves

1. This dish takes a long time since the cornmeal paste needs to cool completely, so make the paste in the morning for an evening meal. 
2. Cornmeal Filling
1. Put the milk in a medium saucepan and slowly bring to a boil, stirring frequently.
2. As the milk warms, add the:
1. salt and pepper
2. nutmeg
3. chives
4. cornmeal and 
5. 3oz of butter. 
3. Stir constantly.
4. When it starts to gently boil, turn down the heat. Use a long-handled wooden spoon and stir stir vigorously to avoid getting burned by the bubbles popping in the thick paste. 
5. Stir until it stiffens to a consistency where a wooden spoon can stand in it. Remove from the heat.
6. Mix in 3oz of the cheese and stir until the batter is smooth.
7. Whisk in the egg until completely incorporated.
8. Line a large cookie sheet with parchment paper and evenly spread the paste to a thickness of 1⁄4" to 1⁄2" and set aside allow plenty of time for it to get cold in the refrigerator, until very stiff.
3. Baking
1. Once the cornmeal filling is cooled, preheat the oven to 350°F.
2. Grease a gratin dish or a lasagna dish. Cut the lump of paste into 1½" diameter rounds or 1½" squares and arrange them in an overlapping pattern like fish scales.
3. Sprinkle the remaining 3oz cheese on top.
4. Put in the oven for 45 minutes to on hour, or until the dish is bubbling nicely all over.
4. Now, turn on the broiler and cook this way until the cheese starts to brown in spots.
5. Browned Butter with Sage
1. Meanwhile, put the remaining butter in a small saucepan and melt at medium-high heat. 
2. The butter will start to froth and making crackling sounds. 
3. When the frothing stops and the crackling slows, look to the bottom of the pot and you will see the milk solids, which have separated and settled there.
4. When the milk solids start to turn golden brown, remove immediately from the heat source - this stage is quite quick when it comes, and left any longer the butter will be in danger of burning.
5. Add the sage leaves and leave for at least 5 minutes for the sage to infuse.
6. When the dish is ready, remove from the oven and pour the browned sage butter over it.
7. Serve immediately as a side or with something a bit tart or vinegary, such as a crunchy salad like Salade de Carrottes, or a selection of pickled vegetables like onions, beets and carrots or Pâtissons Gratinée (summer squash au gratin).