Drain and rinse the hearts of palm, then pat them dry and slice into 1-inch / 2. 5 cm thick rounds.
In a large sauté pan, warm the olive oil over medium-high heat.
Add the hearts of palm in a single layer and sear for 2-3 minutes per side, until golden brown.
Stir in the garlic and cumin, and season with 1 teaspoon (5 ml / 0.2 fl oz) of salt and ½ teaspoon of pepper.
Cook for 30 seconds until fragrant.
Carefully pour in the rum (it may flame briefly), then add the lime juice.
Cook for another minute, allowing the sauce to reduce slightly.
Remove from heat and toss with the fresh mint and 2 tablespoons (30 ml / 1 fl oz) of the cilantro.
Set aside.
In a separate bowl, combine the minced red onion, the remaining 6 tablespoons (90 ml / 3 fl oz) of cilantro, diced avocado, orange segments, and minced jalapeño.
Season with the remaining ½ teaspoon of salt and ½ teaspoon of pepper.
Toss gently to combine.
To serve, spoon the avocado-orange salsa into the center of each plate.
Arrange the seared hearts of palm around the salsa.
Drizzle everything with the mojito pan sauce and garnish with fresh cilantro sprigs.