Crumble the feta cheese and mix it with the yogurt. Chop and crush the garlic cloves, then add them to the cheese and yogurt mixture. Distribute the mixture into deep plates.
In a large pot, pour water and bring it to a boil. Once boiling, add 1 teaspoon of vinegar and ½ teaspoon of salt. Remove the pot from heat and gently crack the eggs one by one into the water.
Return the pot with the eggs to the heat and let them simmer for 2-3 minutes for soft eggs, or 5 minutes for firmer eggs. The eggs should be covered.
Once the eggs are ready, remove them with a slotted spoon and place two eggs in each plate.
In a pan, melt the butter, then add the red pepper and stir for 20-30 seconds, being careful not to burn the pepper as it will ruin the flavor of your Panagyurishki eggs. Add the sautéed red pepper to each serving.