Preheat an outdoor grill or indoor grill pan or griddle to medium-high.
In a small pot over medium, heat 1/3 cup of the EVOO with the garlic and rosemary to infuse, 2 to 3 minutes or until the garlic bubbles.
Arrange a couple of scallions on each escarole quarter and dress with the infused oil and lemon juice; season with salt and pepper. Wrap each greens bundle around its center with the speck, pressing the slices together where they overlap. Brush the bundles with a little EVOO, then grill until the greens' edges are browned and the speck is crisp, about 2 minutes per side.
In a nonstick skillet, melt the butter over medium heat and cook the eggs sunny side up or to desired doneness.
Serve the greens and ham topped with an egg.