For the chutney: Combine the green tomatoes, brown sugar, cider vinegar, curry powder, mustard seeds, fennel pollen, allspice, ginger, apple and red onion in a medium heavy-bottomed pot. Bring to a boil, then reduce to a simmer. Cook, stirring occasionally, for about 30 minutes. Taste and thin with chicken stock as desired. Adjust the seasonings to taste.
For the greens and bacon: Heat the olive oil in a large skillet over medium-high heat. Add the bacon and cook, stirring occasionally, until crisp; remove to a paper-towel-lined plate. Lower the heat to medium, add the garlic and shallot, and cook, stirring occasionally, for 3 to 5 minutes. Add in the chard and beet greens and let wilt for a few minutes. Deglaze the pan with vinegar, then add the sugar, nutmeg, salt and pepper to taste, and a dash of hot sauce. Add some of the bacon back to the pan, and combine. Use the remaining bacon as garnish.
For the butter grits: In a medium saucepan, bring the chicken stock, 1 teaspoon salt and 1 tablespoon of the butter to a boil. Add the grits and bring to a simmer. Stir, then partially cover, and cook, whisking occasionally, until the liquid is absorbed, about 15 minutes.
Add in 1/2 cup milk, whisk and partially cover. Cook, whisking occasionally, for another 15 minutes.
Add in the remaining 1/2 cup milk and cook partially covered, whisking occasionally, another 15 minutes.
Add the remaining tablespoon butter and season to taste with salt. Use additional stock to adjust the consistency as needed. Serve the grits with the quail eggs.