- Once tomatoes have marinated for 1 hour, start preparing leeks. Slice off leek tops where white part becomes dark green and discard (light green is still good; don’t cut that part off). Slice off root at other end, keeping just enough core to hold leek together while cooking. Rinse off any visible dirt with lukewarm water.
- Place medium-sized sauté pan over medium-high heat. Fill large pot with water, cover, and bring to 175 degrees. (If you don’t have a thermometer, look for a little steam, but no simmer).
- Once sauté pan is hot, pour in enough olive oil to coat bottom. Place leeks in pan and sear until lightly brown all around. Smash garlic and add to pan along with thyme. Add butter, and when it starts to foam and brown, slowly pour in wine.
- Bring to simmer and cover, leaving small amount of pan uncovered so liquid can reduce. Turn leeks occasionally for 12-15 minutes, until soft but not mushy. Remove leeks. If braising liquid is not a glaze at this point, continue reducing until a glaze. Discard garlic and herbs, season with salt to taste, and pour glaze over leeks. Set aside an keep warm.
- For eggs, poach in 2 patches, cracking 1 each into 4 small bowls. Stir water in pot to create a circular motion. Drop eggs in 1 at a time and cook 3-4 minutes until whites are set but yolks are not. Carefully remove with slotted spoon and dab with paper towels. Keep warm.
- To assemble, slice each leek in half lengthwise and place on plate. Spoon tomatoes over leeks and add poached egg. Top with slice of ham and drizzle with remaining tomato liquid. Garnish with chives and serve with baguette.