Preheat oven to 450°F (230°C) with rack in upper-middle position. Line a rimmed baking sheet with parchment paper and lightly coat with cooking spray or oil.
Cook bacon in a skillet over medium heat until crispy, about 6-8 minutes. Drain on paper towels, cool, then crumble into small bits.
Separate eggs one at a time, placing each yolk in its own small bowl and adding whites to a large mixing bowl. Ensure no yolk gets into the whites.
Add salt to egg whites and beat with an electric mixer on low speed until foamy, then increase to medium-high and whip until stiff, glossy peaks form, about 2-3 minutes.
Gently fold in crumbled bacon, Parmesan, chives, and black pepper using a rubber spatula. Fold just until combined, about 6-8 folds, being careful not to deflate the whites.
Drop 4 large mounds of the mixture onto the prepared baking sheet, spaced 2 inches apart. Shape into nests with a shallow indentation in each center.
Bake for 3 minutes until the whites are set and just beginning to brown at the tips. Remove from oven and re-create the indentations if needed.
Carefully slide a yolk from its bowl into the well of each cloud. Return to oven and bake 2-3 minutes longer, until yolks are runny to jammy as desired.
Remove from oven and sprinkle with flaky sea salt, extra pepper, and chives. Serve immediately on toast or alone while the whites are still puffy.